he has placed a bet for $10,000
It is not the "continue" gambling meaning in the WR dictionary ("let the bet ride" to the next round of gambling). It is a single bet.
"I've got ten big ones riding on the game" means "I have wagered $10,000 on the outcome of the game."
In your original question, the quote refers to a video game. It seems unlikely that someone would place such a large bet on the outcome of a video game (though it's possible) and we don't really know whether he's playing the game or watching someone else play it. It is possible that he actually means that he's invested $10,000 in the development of the game, and he stands to lose his investment if the game does not become popular. We don't know from the small amount of context provided, and I am not familiar with the show.