I have been to some interesting places in my own brain with the help of psychedelics. While I have experienced different levels of self-awareness, the possibility of that "filling up my brain" is likely not possible.
There is a part of the brain that psychedelics specifically affect that functions as kind of a traffic regulator. It typically only directs signals from one part of the brain to another part. Psychedelics open these pathways up and allow for information to flow to in all kinds of directions. (Synesthesia, sensory confusion, is an example: Feeling colors and seeing sounds.)
What the experience does for me, is that I seem to gain more awareness into how my brain works. It's like I can stand back and watch how my brain processes things. My subconscious is pulled into full view and I can almost tinker with it, in a way. Some theories suggest that the colors and geometric patterns people tend to see is our actual brain operations being leaked into the visual cortex and/or the fractal patterns are the result of actual "data loops" caused by psychedelics allowing information to pass around freely. (Not my theories, btw.)
Now, you may read this and think: That dude is just tripping! (And you would be very much correct.) The thing is, every experience I have and anything that I feel is already in my brain. The data is already there, but how that data is processed is vastly different. Even if I am perceiving parts of my brain that I couldn't before, it's still just the same neurons that were always there.
So, what I am basically saying is, is that my self-awareness temporarily becomes self-aware. It's a shitty description, but it's the closest I could get to matching the situation you were asking about.
I still develop memories, almost as normal. Some memories stick and some fade. All that really happened is that a few neuron weights got shaken up and it all becomes a few new pathways with a similar number of neurons as before. (Neurogenesis and dendritic growth as a result of psychedelics is a different subject and I wouldn't think it would be part of the recursion-type situation you are asking about.)
Memories become integrated with existing ones, basically. While vastly different in many ways, our current AI tech has a set number of "neurons" it can work with in any given model. You can train those same bunches of neurons with millions of different patterns, but the actual size of the neural network doesn't change, no matter how much data you cram into it. With slight changes to when neurons fire, you are using specific pathways of neurons to save data, not necessarily using the neurons themselves.