Is there a record for most stones to fill the board in a go game? Dumb question I know, but its something I've been curious about for awhile. This game left very little real estate on the board at its conclusion

Two comments:

I am not sure exactly what you are asking. Is it "Is there a record for the fewest vacant points at the end of a game?"?
The position you show is not at the conclusion of a game, there are still unfilled dame points. If you were using Chinese rules, it was in the players' interests to fill these; if Japanese, the rules require you to fill them, though they do not absolutely insist on it.

Are you talking about a real game? Because it could go all the way to 359 if the players are willing...

The longest pro game is 400 and something, I don't remember where I saw it.

There was a game with 411 moves between Yamabe Toshiro and Hoshino Toshi in the Oteai on 1950-12-20 listed at FamousGamesInvolvingKo.

Of course this involves ko, so it wouldn't necessarily be the game ending with the least open spaces at the end.

The OP has a good question: what's the pro game that has ended with the least open spaces at the end? Or perhaps: what's the pro game with the least open spaces after adding in dame if the game was scored with japanese scoring. Or: what's the pro game with the least open spaces at the point at which the temperature dropped to zero?

Technically this is after the dame are filled because unless the dame are filled, all stones touching them are in seki.

That's the one. Thanks prokofiev. Though if there are real dan or even high kyu games that beat the pro record it would be worth seeing those as well no?