We all know that these titles are not "given" but are descriptions of the state of consciousness of the individual. I'll tell you my understanding based on Vedic Science :
"Rishi" means seer. But what does a Rishi see? The fine fabrics of consciousness. You might have heard of "sruti" which means "that which is heard". The Veda, or knowledge is not written in a book. The Veda is cognized in the awareness of an individual. It is both heard and seen on a deep level of consciousness. Just like a physicist explores particles and frequencies in Nature, so also a Rishi explores the frequencies and fluctuations of conscious. So a Rishi hears and sees the sounds/impulses of the Veda (spandan), and understands that the sounds manifest into material forms around us. The Rishis of the Vedic literature all cognized the Veda in their own awareness. If you are to hear these sounds, or see the impulses, you too have experienced that particular Rishi consciousness. This is why different aspects of the Veda have different Rishis, different Devata and different chhandas according to the Cognition. Rishis have cognized the 40 aspects of the Vedic literature which deal with everything from vastu, Ayurved to Brahmi Chetan. Patanjali, who cognized Yog shastra, attained the state of Yog intelligence and he saw the impulses in his own awareness. Anyone who sees from that level has achieved "Patanjali Rishi consciousness". The Rishi name is expressed in the form of their Cognition (nama-rupa).
One thing is to know, see and hear the Veda, but another thing is to do something with that knowledge. A Maharishi is one who takes that vision and knowledge of Natural Law and can create transformation in the world. Maharishis are teachers and Vedic engineers who use techniques to transform their students (and the world) by putting the principles of consciousness into practice. Even their Conscousness is catalytic in this transformation.
For this reason the higher one is in Conscousness, the greater their ability to administrate life around them. A Raja (or king) normally administrates human life, but as one goes in consciousness, one is able to administrate cosmic life. This is why sages are called Maharaja.
A Brahmarishi is one operating from the level of Brahm who is transforming... and therfore administrating from the level of the universe. This is the reason Brahmarishis are considered to be administrators of the universe. Their Conscousness is operating from that level of intelligence which manages the universe.
A Devarishi (as a commentator mentioned above) is one who has cognized Brahm (the Totality) ... And into that level of consciousness from which all universes have originated, but also play a role in being a teacher to the devas. This is why (for example) there have been descriptions of Narada, that when he journeys here and there he comes from a place where all of the universes are administered. The Devas are impulses of Natural Law which transcend this world/universe and the next.
From my understanding (but I am not certain about this) a rajarishi is one who is a king on earth who is also a Rishi. I know from the Vedic literature that Rishi Rajas such as Janaka were given this title.
There are many more levels that are not mentioned in this question : such as Sidhas and Naths, etc. Unfortunately, many people just use these terms to show respect, but actually they are terms with specific meanings.
Once one knows this, one desires to attain this level of consciousness! Begin by practicing meditation every day and unfold the knowledge from within.