That depends entirely on the person playing the game.
I don't think I could do literally hundreds of games, but I was up to around 17 a little while ago and didn't have too much difficulty remembering my previous analysis and ideas.
One thing I have taken do doing though is instead of using the Analysis board, I'll do it using software (not an engine, just software). That way not only can I annotate my analysis and all the variations I checked, but I can save it as a PGN. Later on, if for whatever reason (aka I'm playing drunk again) I can't remember what I thought about a specific position, I can always go look it up.
True story... I started doing this just recently after blundering a pawn-up rook-pawn endgame I was trying to win while intoxicated .
There seems to be a few camps on chess.com with regard to the number of concurrent games; those that enjoy a few games at a time, players that have dozens or hundreds going at once, and the middlers, say 20 or so. It seems that those at the ends of the spectrum are playing two different types of chess: A complete cohesive game where prior moves are in memory and at least resembles OTB, and chess puzzles (unless you have photographic memory).
It seems like the site could benefit by enabling a player to further restrict opponents in challenges and tourneys by selecting maximum games in progress. Not only would players with similar motive and style find each other, tourneys could move along to completion while still giving the flexibility of 3 dpm.