Is there a version of Pokemon in which you could get the leftover starter?

There is no game where you can claim the leftover Pokemon after choosing your starter.

In some games (such as Black and White), both unchosen Pokemon are given to NPCs. In others (like Red and Blue), the unused Pokemon sits neglected on a table all game long. There is no way for the player to get this Pokemon in any version.

In Pokemon Yellow, it is possible to get all three first-generation starters, although you don't start with any of them, so that still doesn't meet the criteria you set forth.

In some games, it is possible to get starters from past generations by completing certain requirements after beating the elite four. For example, in HeartGold and SoulSilver, you can get a 3rd-generation starter in the Silph Co. building, but you can never pick up the "missing" 2nd-generation starter that wasn't picked early on in the game.

You might be thinking of a trading cheat used in Red and Blue to get all the starters with only two games and not having to do much work.

Once you start a game you could get your starter, get to the point where you could use pokeballs and the pokemon trading center. There you could trade with another person who wanted the starter pokemon for something like a rattata or pidgey. Then you would reset the game, and start the process over again to get a different or more starters. Then when you are done, trade the pokemon back to your game to get 'ALL' the starters.

I remember this because all of my friends had gotten to the fourth gym, but didn't want to trade their starters away or reset their game. Since I just started, my game was the 'reset' game everyone could use to get starters because I hadn't gotten too far.