Beginning last month, some of my games are not in the 'My Games' section of the Opening Explorer. Is there a limit to the number of games this software will hold?
No, there is not any limit that I have ever heard of.
How many are missing?
I saw 3,817 games as white out of 7,882 total, so it would seem that most are there.
So far, I note that these games are missing, many recent daily games with black pieces:
https://www.chess.com/analysis/game/daily/281915208?tab=analysis
https://www.chess.com/analysis/game/daily/272969006?tab=analysis
https://www.chess.com/analysis/game/daily/274736346?tab=analysis
https://www.chess.com/game/daily/273997218
https://www.chess.com/analysis/game/daily/279173068?tab=analysis
https://www.chess.com/analysis/game/daily/279173068?tab=analysis
Double that, It's annoying how this issue pops again and again, for me and other players as well.
Wanted to prepare against my friends, but none of their games is listed in opening explorer (and they played quite a dozen of games).
Googled the same from 2014, it's weird how chess.com just can't fix their stuff 
UPD: and from 2009 
my game is rigged
Beginning last month, some of my games are not in the 'My Games' section of the Opening Explorer. Is there a limit to the number of games this software will hold?