Is there a similar hierarchy for Devi's incarnations? If yes, what is that hierarchy?
Before knowing about hierarchy of incarnation of Devi, it is necessary to know about hierachy of Devi herself (with respect to other Gods). So, to explain this I have divided the answer to this question to several parts:
1) Explanation from Absolute Level:
Below is the explanation from the absolute level point of view or Paramartha Satyam;
● Devi as only one that exists:
It is Devi only that exists in the entire universe. No second other than Devi exists.
In the SaptaShati (Markandeya Purana) before killing Sumbha Devi states:
एकैवाहं जगत्यत्र द्वितीया का ममापरा।
पश्यैता दुष्ट मय्येव विशन्त्यो मद्विभूतयः॥५॥
I only exist in this entire Universe. There is no one besides me. Oh wicked one it is myself who is created in many forms.
This clearly shows there is no second than Devi. Similarly in Shiva Purana chapter Manifestation of Uma, Devi states:<
परं ब्रह्म परं ज्योति प्रणवद्वन्द्वरुपिणी ।
अहमेवास्मि सकलं मदन्यो नास्ति कश्चन ।।
I am the Supreme Brahman the great lusture in two Pranavas. I am everything that exist, nothing exists besides me.
यथा दारुमयीं योषां नर्त यत्यैन्द्रजालिकः ।
तथैव सर्वभूतानि नर्तयाम्यहमीश्वरी ।।
Just as a magician makes a wodden puppet to dance, in the same way I make dance the entire beings.
Thus the above verses clearly show that there is no second besides Devi that exists. ie. Everything is Shakti herself.
● Devi as Purusha:
Most people only think Devi as Pradhana and thus conclude Devi can't be Brahman as Brahma Sutra reject the eternity of Pradhana. While the truth is Devi is Purusha herself too!
The Purush Sukta of Rig Veda 10.90 states:
पुरुष एवेदं सर्वं यद भूतं यच्च भव्यम |
This Puruṣa is all that yet hath been and all that is to be.
ie. The necessary attribute of Purusha is it is all that was that is and that will be. The same RigVeda in 10.89.10 states:
अदितिर्द्यौरदितिरन्तरिक्षमदितिर्माता स पिता स पुत्रः |
विश्वे देवा अदितिः पञ्च जना अदितिर्जातमदितिर्जनित्वम || (RigVeda 10.89.10)
Aditi is the heaven, Aditi is mid-air, Aditi is the Mother and the Sire and Son. Aditi is all Gods, Aditi five-classed men, Aditi is all that hath been born and shall be born.
So, this clearly shows that Aditi/Devi herself is the Purusha.
Similarly the first verse of Purush Suktam of Rig Veda states:
सहस्रशीर्षा पुरुषः सहस्राक्षः सहस्रपात |
A THOUSAND heads hath Puruṣa, a thousand eyes, a thousand feet.
And the Lalita Sahasranaama which was told by Lord Hayagriva to Maharshi Agastya in Brahmanda Purana (starting from the 282th name states):
Having thousand headed body, thousand eyes and thousand legs.
Thus it is clear Devi is not only Pradhana but also Purusha.
●) Devi as Brahman and Brahma-Vidya:
Devi is both Brahman and Brahma Vidya. In the Kena Upanishad Devi is in form of Brahma Vidya:
III-12. In that space itself (where the Yaksha had disappeared) Indra approached an exceedingly charming woman. To that Uma, the daughter of the Himalayas, he said: “Who is this Yaksha?”
IV-1. She said: “It was Brahman. In the victory that was Brahman’s you were revelling in joy”. Then alone did Indra know for certain that It was Brahman.
As Devi gave knowledge of Brahman to the Gods above, so Devi here is in the form of Brahma Vidya. In the Katha Upanishad we have:
2-I-7. (He who perceives) this Aditi that comes into being as the Prana, comprising all the gods, that is manifested along with the elements, and that, entering into the cavity of the heart, is seated there, he perceives that very Brahman. This verily is that (thou seekest).
Here Aditi/Devi is herself Brahman.
So as a conclusion from explanation from Absolute level; Devi is herself Purusha, Devi herself is all that exists; Devi herself is Brahman and Brahma Vidya. So as everything is Devi from absolute point of view there is nothing such as Hierachy of Incarnation of Devi.
2) Explanation from Relative (Vyavaharika level):
The world which we perceive now is full of duality (due to our ignorance), so the explanations from relative level is also needed so that it helps to understand absolute level.
● Devi as the innate power of all Gods:
To such Rudra who is glorified as Brahman in the Shewatashwatara and Atharvasiras Upanishad, actually the innate power of Rudra is Devi herself. In the Devi Suktam of RigVeda Devi states:
अहं रुद्राय धनुरा तनोमि ब्रह्मद्विषे शरवे हन्तवा उ।
अहं जनाय समदं कृणोम्यहंद्यावापृथिवीआविवेश॥६॥ (RigVeda 10.125)
I bend the bow of the Rudra to kill all those enemies who detest all good things. I fight these bad elements / enemies only for the people. I enter, pervade and persist throughout the earth and the sky.
So, it clearly shows that Devi is the innate power and actually it is her who bends bow of Rudra.
Similarly it is Vaishnavi Shakti who remains as innate Shakti of Vishnu and performs all tasks (which people say performed by Vishnu). In the SaptaShati during Narayani stuti it is stated:
प्रसीद वैष्णवीरूपे नारायणि नमोऽस्तु ते॥१६॥
I salute Narayani, the possessor of the
Sanka, Chakra, Gada, and Sargna; who favors all in the form of Vaishnavi.
Similarly it is Maheswari Shakti which makes all work to perform for Maheswara:
माहेश्वरीस्वरूपेण नारायणि नमोऽस्तु ते॥१४॥
I salute Narayani, holder of the Trishula, Chandra, who rides on the ample ox in the form of Maheswari.
Actually it is Aindri Shakti which is innate power inside Indra who killed Vrittasura (and people due to ignorance assume it is Indra who killed Vritta):
किरीटिनि महावज्रे सहस्रनयनोज्ज्वले।
वृत्रप्राणहरे चैन्द्रि नारायणि नमोऽस्तु ते॥१९॥
I salute Narayani, who is crowned, the holder of the Vajra, who glowes with one thousand eyes, the drawer of the soul of Vritra in the form of Aindri.
So it is actually innate power inside gods which perform every task. Eg. Maheswara didn't destroyed Tripura, it is Maheswari Shakti which is inside Maheswara who did that task. It is not Varaha who lifted the earth, it is Varahi Shakti inside Varaha who did that. The first verse of RigVeda Devi Suktam itself states:
अहं रुद्रेभिर्वसुभिश्चराम्यहमादित्यैरुतविश्वदेवैः |
अहं मित्रावरुणोभा बिभर्म्यहमिन्द्राग्नीहमश्विनोभा ||
I TRAVEL with the Rudras and the Vasus, with the Ādityas and All-Gods I wander. I hold aloft both Varuṇa and Mitra, Indra and Agni, and the Pair of Aśvins.
So, it is Devi herself who is innate power of all Gods and travel with all Gods.
●) Devi incomprehensible even to Hari and Hara:
In the SaptaShati Devas pray to Devi as:
ब्रह्मा हरश्च न हि वक्तुमलं बलं च।
नाशाय चाशुभभयस्य मतिं करोतु॥४॥
Thy miracles are unparalleled, Bhagavan, Ananta, Brahma, and Hara are unable to express it. Mayest thou, O Chandika! endeavour to preserve the world, and dispel the fears of the giants.
In the Shiva Purana chapter manifestation of Devi, Devi herself state:
न ब्रह्मा न मुरारातिर्न पुरारातिरीश्वरः ।
मदग्रे गर्वितुं किंचित्का कथान्यसुपर्ववाम् ।।
Neither Brahma Nor Vishnu nor Shiva can boast before me, what to speak of other Gods.
●) Devi herself as the performer of creation, protection and destruction:
In the last verse of RigVeda Devi Suktam, Devi states:
अहमेव वात इव प्रवाम्यारभमाणा भुवनानि विश्वा।
परो दिवा पर एना पृथिव्यैतावती महिना संबभूव॥८॥ (RigVeda 10.125)
I breathe a strong breath like the wind and tempest, the while I hold together all existence. Beyond this wide earth and beyond the heavens I have become so mighty in my grandeur.
Thus Devi herself is the support of all this existence. In the Narayani Stuti in Saptashati Gods pray Devi as:
सृष्टिस्थितिविनाशानां शक्तिभूते सनातनि।
गुणाश्रये गुणमये नारायणि नमोऽस्तु ते॥११॥
I salute Narayani who is the eternal power for creation, preservation and destruction. Who is without attribute and full of attribute.
Thus, Devi is the eternal power of Sristi, Sthiti and Vinasha. It is supported by Vedas also as already discussed. In the SaptaShati Brahma prays to Devi as:
त्वयैतद्धार्यते विश्वं त्वयैतत्सृज्यते जगत्॥७५॥
त्वयैतत्पाल्यते देवि त्वमत्स्यन्ते च सर्वदा।
विसृष्टौ सृष्टिरूपा त्वं स्थितिरूपा च पालने॥७६॥
तथा संहृतिरूपान्ते जगतोऽस्य जगन्मये।
Thou art the sustainer of all things, thou hast created all things, thou governest all things, thou existest to the end constantly. Thou art the principle of creation, thou art plastic, thou art the preserver, thou art
the destroyer in the end, thou pervadest through the universe.
There are other many verses which state it, in the SaptaShati.
●) Devi herself in the form of Brahma, Vishnu and Rudra:
From above we know that it is Devi who is cause behind the creation, protection and destruction. Not only this Devi herself is in the form of Brahma, Vishnu and Rudra also who perform this task. As already stated in RigVeda "Aditi is all Gods." Similar things come here:
In the chapter Manifestation of Uma from Shiva Purana Devi herself states:
विरञ्चि सृष्टिकर्ताऽहं जगत्पाताऽहमच्युत ।
रुद्रः संहारकर्ताऽहं सर्वाविश्वविमोहिनी ।।
I am the creator Brahma, protector Achyuta and destroyer Rudra and also enchater of the whole world.
And similarly the 264th name of Devi in Lalita Sahasranaama is given as:
सृष्टीकर्ता ब्रह्मारुपा ।
who takes the form of Brahma for creation.
Similarly 265th name is:
गोप्त्री गोविन्दरुपिणी ।
who protects in the form of Govinda (Vishnu).
Similarly 266th name is:
संहारीणि रुद्ररुपा ।
who destroys in the form of Rudra.
Thus, it is Devi Herself who is in the form of Brahma, Vishnu and Rudra.
Thus in the Relative level also everything is seen as just forms of Devi which was already established in Absolute level.
3) Establishing Hierachy in Relative level:
Although everything is seen as forms of Devi in Relative (Vyavaharika) level also. Still we can establish Hierarchy in forms as Relative level is subject to Naama-Rupa (Name- Form) and other things like Time, Maya, Law of Karma etc..
● Supreme form of Devi:
In the Tripura Rahasya chapter 20 "Vidya Gita" Devi herself states as:
My concrete form is the eternal couple — the Supreme Lord and Energy — always in undivided union and abiding as the eternal consciousness pervading the three phenomenal states of waking, dream and sleep, and reclining on the cot, whose four legs are Brahma (the Creator), Vishnu (the Protector), Shiva (the Destroyer) and Ishwara (Disappearance) and whose surface is Sadasiva (grace), which is contained in the mansion known as the ‘fulfilment of purpose’, enclosed by the garden of ‘kadamba’ trees, in the jewel island situated in the wide ocean of nectar surrounding the cosmos and extending beyond.
Thus the Supreme form of Devi (Sri Lalita TripuraSundari) is residing in cot whose four legs are Brahma, Vishnu, Rudra and Maheswara. And Surface is SadaShiva. ie. She is residing above SadaShiva also.
Though Sri Lalita is chaste wife of SadaShiva. LalitaSahasranama states:
सदाशिव पतिव्रता ।
who is chaste wife of SadaShiva.
Brahma, Vishnu, Rudra, Maheswara and SadaShiva are known as PanchaBrahmanas and detailed explanation of it is avaliable in Linga Purana and Shiva Purana. Sri Lalita is higher even than these PanchaBrahmanas:
58th name of Lalita Sahasranaama states:
who is seated in Five Brahamanas (Brahma, Vishnu, Rudra, Maheswara, SadaShiva)
249th name of Lalita Sahasranaama states:
who is seated in five corpse.
Here these Five are actually PanchaBrahmanas and they are compared to corpse because PanchaBrahmana (Brahma, Vishnu, Rudra, Maheswara and SadaShiva) are like corpse without power of Devi.
250th name states:
who is in the form of PanchaBrahmanas.
Thus Devi herself is in the form of PanchaBrahmanas also.
One of the name state:
who is the witness of MahaTandava performed by Maheswara at the end of MahaKalpa.
Thus her form only remain at the end of 100 yrs. of Brahma (Mahakalpa); so she is the witness of MahaTandava performed by Maheswara at the end of MahaKalpa.
Similarly, 281th name states:
who creates and destroys world with the blinking of her eyes.
And similarly there are other verses also. The name Lalita itself means "who does Lila" or who plays. Thus she is verily Brahman herself and she is highest in the Hierachy of Devi.
●) Forms of Mahakali, Mahalakshmi and Mahasaraswati:
Generally there is a misconception that Mahakali, MahaLakshmi and Mahasaraswati denote to consort of Rudra, Vishnu and Brahma respectively.
However, the term MahaKali, MahaLakshmi and Mahasaraswati denote the Amsas of Supreme Devi presiding over Tamasic, Rajasic and Satvik mode respectively.
For instance MahaKali kills Madhu Kaitabha and MahaLakshmi is invoked to kill Mahisasura. However this MahaLakshmi is not wife of Lord Vishnu. Similarly Sumbha Nisumbha is killed by MahaSaraswati but she is not wife of Brahma.
For clearing confusion the chapter 48 of Uma Samhita Shiva Purana discuss about manifestation of Mahakali as:
ब्रह्मणा प्रार्थिता सेयं मधुकैटभनाशने ।
महाविद्या जगद्धात्री सर्वविद्याधिदेवता ।।
द्वादश्यां फाल्गुनस्थैव शुक्लायां समभून्नृप ।
महाकालीति विख्यितां शक्तिस्त्रैलोक्यमोहिनी ।।
On the prayer of Brahma, for the destruction of Madhu Kaitabha; Mahavidya, Jagaddhatri, the great Godesses of all the Gods, on the 12th day of the bright fortnight of the month of Falguna, possessing the power to delude all the Lokas, known as Mahakali emerged who became well known in the world.
Similarly chapter 46 talk about invocation of MahaLakshmi while killing Mahisasura:
ततोऽम्बिकां परां शक्तिं महालक्ष्मीस्वरुपिणीम् ।
तुष्टुवुस्ते सुराः सर्वे भक्तिगद्गदया गिरा ।।
The Gods then prayed to the Supreme Godesses Ambika in the form of MahaLakshmi with choked voice in devotion.
एवं ते कथितो राजन्महालक्ष्म्याः समुद्भवः ।
सरस्वत्यास्त्तथोत्त्पत्तिं श्रृणु सुस्थेन चेतसा ।।
O King in this way the origin of MahaLakshmi has been spoken to you. Now with a cool mind listen to origin of (Maha)Saraswati.
Similarly chapter 48 discuss about "Manifestation of MahaSaraswati" while killing Sumbha Nisumbha:
एवं देवी समुत्पन्ना शुम्भासुरनिवर्हिणी ।
प्रोक्ता सरस्वती साक्षीदुमांशाविर्भवानृप ।।
This is how the Godesses who destroyed Sumbhasura was born. She was born out of the amsa of Devi and is called (Maha)Saraswati.
Thus Mahakali, MahaSaraswati and MahaLakshmi doesn't denote to wife of Rudra, Brahma and Vishnu here. They are the presiding powers over mode of nature as Tamasic, Satvik and Rajasik respectively.
●) Forms of Kali, Lakshmi and Saraswati:
By using the term, 'Kali, Lakshmi and Saraswati' I'm referring to wife of Rudra, Vishnu and Brahma respectively.
As already discussed every forms are form of Devi and so is the form of Parvati, Lakshmi and Sarawati also.
In the Chapter 49 Uma Samhita of Shiva Purana, Devi states:
कालिकाकमलावाणीमुखाः सर्वा हि शक्तयः ।
मंदशदिव संजातास्तथेमा: सकला: कला ।।
Shaktis like Kalika, Lakshmi Saraswati and all the other Shaktis also emerge out of my body. All the arts also orginate from me.
Similarly while praying to Durga, Brahma in Rudra Samhita Shiva Purana ( chapter 11) states:
यथा धृतशरीरा त्वं लक्ष्मीरुपेण केशवम् ।
आमोदयसि विश्वस्य हितायैतं तथा कुरु ।।
Just as you assuming the form of Lakshmi, pleased Vishnu, similarly for the benefit of the Universe, you also do this.
So, it is Devi herself who incarnates as Lakshmi and Parvati and Saraswati.
However SadShiva being consort of Supreme Devi and Rudra being direct incarnation of SadaShiva, so Supreme Devi also directly incarnates as Sati and Parvati and become spouse of Shiva. So, by the term Devi we generally understand incarnations of Sati and Parvati. This is the reason why Parvati is more emphasized in Saptashati and this is the reason why Rudra Parvati relationship is explained with concepts of Shiva Shakti. So, in the hierachy level we can keep Parvati/Sati before Lakshmi and Saraswati as she is direct incarnation (from Shakti point of view).
All other Shaktis of other Gods also come in similar way. In this way same Devi incarnates in various forms and become Shakti of all! In any case all these incarnation, hiearchy thing are only applicable in Vyavaharika level that too due to ignorance of Naama-Rupa (Name Form).
Thus as a conclusion from above we can say:
1) Everything that exists is Devi from absolute point of view.
2) Everything are forms of Devi from relative point of view also.
3) Under the Naama-Rupa describing Supreme form of Devi in Relative level she is Sri Lalita TripuraSundari.
4) MahaSaraswati, MahaLakshmi and Mahakali are Amshas of Supreme Devi and they preside over Satva, Rajasa and Tamasa attribute.
5) Parvati, Lakshmi, Saraswati and all other Shaktis are incarnation of Supreme Devi.
6) All these Hierarchies are seen due to ignorance under Nãma-Rupa (Name-Form).😀😀
Such a hierarchy is detailed in the Devi Bhagavata Purana. The five major forms of devi are :-
Durga, Lakshmi, Saraswati, Savitri(aka Gayatri) and Radha.
"The ‘Mula Prakriti’ which emerged thus manifested into five major Forms, viz. Durga, Radha, Laxmi, Sarasvati and Savitri."
See http://www.kamakoti.org/kamakoti/details/devibhagvatpurana65.html for a summarised note on the hierarchical manifestations of Devi.
MahaLakshmi = Adi ParA Shakti is the supreme one and she becomes the various Goddesses.
MahaLakshmi "created" the other 2 mahas and the tridev-pairs as described in Rahasya Thrayam 1 - Pradhanika Rahasyam as follows:
Supreme Goddess MahaLakshmi saw emptiness and hence took up tAmas to manifest KAli-form. Also known as Mahamaya MahaKAli, Mahamari, KshudhA, Nidra, KAlarAtri
Mahalakshmi then manifested MahaSaraswati with beads and ViNa musical instrument and gave her names like VANi, Vak Saraswati, Bhadra
MahaLakshmi instructed MahaKAli and MahaSaraswati to create the Tridev brother-sister pairs i.e. KamalA-BramhA (lotus, red-pink, creative-oppulent), Shiv-Saraswati (white, knowledge), Krishna-Kali (dark, power-vAtsalya-sustenance-intelligence), and pair them - BramhA Saraswati, Shri-VAsudev, Shiv-Gauri.
Note that DurgA is a manifestation of MahA Lakshmi at a later point in time for a purpose - to kill MahishAsur. DurgA = Result of collective energy of all Devas as initiated by BhagvAn VishNu/MahaVishNu.
Please see my this answer post that explains the secret answer to the treasure hunt called Devi in detail.