Summary: It would be more appropriate to ask the Ramayana of how many Kalpas is available (explained in 1). As discussed (in 2), various scriptures sink into oblivion causing them to be written again, especially the itihasas. This means the Ramayana of many Kalpas is almost impossible to find. However in some surviving scriptures, the details of former Kalpas have been given, which I have enumerated (in Point 3).
1. Frequency of the Ramayana
First let’s see, that as discussed in the chronology of Bhagavan Vishnu’s avatars, the Ramayana we are aware about, has occurred in the present Vaivasvata Manvantara in the 24th Treta Yuga only, not in the 28th Treta yuga. It was penned down by Valmiki, who was the Vyasa of the 24th Mahayuga. The 28th Mahayuga on the other hand was blessed by the Krishnavatara. Thus there is only one Ramayana currently for the present Manvantara. No different Ramayana for different Mahayugas.
From the chronology of the avataras one can see that the Ramayana doesn’t repeat in Mahayugas, but rather, as Tulsidasji says, in Kalpas:
नाना भाँति राम अवतारा। रामायन सत कोटि अपारा॥
कलपभेद हरिचरित सुहाए। भाँति अनेक मुनीसन्ह गाए॥
There are many Ramavataras and a hundred crore (countless) Ramayanas. Owing to difference in Kalpas, the Rishis have sung the glories of Hari in different styles.
-Shri Ramcharitmanas Balakanda Doha 33.3-4
Accordingly a more appropriate question would be - the Ramayana of how many Kalpas is available?
2. The Ramayana of How Many Kalpas is available?
As discussed here, verse 30 of Chapter 22 of the Nirvana Prakarana of the Yoga Vasishtha says that Kakabhushundi remembers Shri Rama descending on earth 11 times. He also mentions that the itihasas sink into oblivion after being written in every creation (verse 25 for Maharamayana [Yoga Vasishtha not Ramayana story] and verse 26 for Mahabharata). Obviously this is the main reason for re-writing them. Accordingly it will be very difficult to get much information about Ramayana of the former Kalpas occurring 11 times and the main focus is towards the current kalpa only.
3. Detailing some information about Ramayana of other Kalpas
However, some light on this aspect has often been shed in various surviving texts.
- The Ramayana of a former Kalpa is narrated in Chapter 116 of the Patala Khanda of the Padma Purāṇa which has a few differences.
- The best description (as per me) of the reason for the avatars in various Kalpas is given in the Balakanda of the Ramcharitmanas by Tulsidasji. He discusses the reason of birth in 5 Kalpas. Briefly they are:
एक कलप एहि बिधि अवतारा। चरित पवित्र किए संसारा॥2॥ - In one kalpa the avatara was in this manner who made the world pure with his character.
-RCM Balakand 123.2
- Once Jalandhara was cursed to become Ravana
एक कलप सुर देखि दुखारे। समर जलंधर सन सब हारे॥ - In one Kalpa seeing the devatas aggrieved after having lost to Jalandhara in battle...
-RCM Balakanda 123.3
- The curse of Narada on Shri Hari (RCM Balakanda Doha 124-139) as described in the Shiva Purana Rudra saṃhitā, Sṛṣṭi khaṇḍa, Chapter 1-4:
- O inflictor of miseries upon others, you shall get the misery of separation from a woman. You shall have the travails of a human being deluded by ignorance.”
-(Shiva Purāṇa) Chapter 4 verse 17
एक कलप एहि हेतु प्रभु लीन्ह मनुज अवतार। - In one Kalpa Prabhu took an avatara for this reason
-RCM Balakanda Doha 139
- The Ramavatara which Sati had encountered while testing Shri Rama took place due to the penance of Swayambhuva Manu and Shatarupa (RCM Balakanda Doha 141-152)
- Lastly the story of Pratapabhanu, king of Kaikeya. (RCM Balakanda Doha 153-175)
- Another slightly different version (from the perspective of Rama’s tirth yatra) is found in the Ananda Ramayana Purna Kanda, 7th Sarga. Here the student, Vishnudasa questions the teacher, Ramadasa as to why his version of Ramayana is a little different. To this Ramadasa replies:
पुनः पुनः कल्पभेदाज्जाताः श्रीराघवस्य च। अवतारा कोटिशोऽत्र तेषु भेदः क्वचित्क्वचित्॥२९॥ कृतोऽस्ति राघवेणैव न सर्वे सदृशाः कृताः।
Owing to Kalpabheda, the Ramavatara has happened again and again many times and there are a few differences in them. Despite the fact that Rama only has conducted these pastimes, they’re not alike in the various versions.