Outcomes:

1 flip --- $1 probability 1/2
2 flips -- $2 probability 1/4
3 flips -- $4 probability 1/8
4 flips -- $8 probability 1/16
...

The expected value doesn't converge but it grows extremely slowly, where almost all the benefit comes from an extremely tiny chance of extremely large gain. The obvious question is counterparty risk: how much do you trust the person offering the game to actually be able to follow through with what they offered?

If we think of this as a sum over coin flips, each flip you think is possible gives another $0.50 in expected value. So if you think they're probably only good for amounts up to $1M then because it takes 20 flips to pass $1M the expected value is $0.50 * 19 or $9.50. Similarly if you think they're good for $1B then that's 29 flips max for an expected value of $14.50. You could be fancy and try to model your uncertainty about how much they're good for, but that's probably not worth it. And you do want to take into account that someone offering something like this with no provision for how they'll handle extremely large payouts is probably not entirely on the level.

Expected value is also not the right metric here, since we all have diminishing marginal returns. Would you enjoy $1B 1,000x as much as $1M? Even if you're giving your winnings to charity there are still some limits to our ability to effectively use additional donations.

Short answer: $5. (This trusts them to be good for $1024, and is in a range where utility should still be pretty much linear in money.)