Statistics/probability question when events are not independent of each other

Let say you play a coin flipping game where a coin is flipped 7 times in a row. In the rare event that the coin lands on heads (H) all 7 times, you win; otherwise, you lose. Each game of 7 flips is independent of any other game of 7 flips. Thus, even if you lose a game by flipping a tail followed by 6 heads, in your next game you must still flip 7 heads. Thus, your chances of winning a single game is

0.5^7 = 0.0078125.

Interestingly enough, if you play that game 90 times in a row, you are more likely to have had this "rare" event happen and win than not. The odds of winning at least one game are

1 - ((1-0.0078125)^90) = .506 = 50.6%

When we play those 90 games, we flip the coin a total of 630 times (90*7). How do the odds work out if we just flip the coin 630 times looking for 7 heads in a row? Is there any way to compute that without some unwieldy conditional computation that carries out for 630 terms or so? Maybe something from calculus that would give us a limit at 630 flips or some other equation?

Thanks for any help on this. My brain has officially shut down.