This is an interesting question and actually a lot more complex than you probably think.

The answer is that it depends on your initial estimate of the probability that the coin is fair, the coin is 55%, etc.

If you initially decide the chances are 100% that the coin is fair, a million trials would not be enough to convince otherwise. If you initially decide that the chances are 1 in a million that the coin is biased towards 55% heads, and otherwise fair, it turns out that after your 1000 trials the chances are about 22.7% that the coin is fair.

If you estimate 1 in a 1000 chance the coin is biased towards 55% heads, the chances are well less than 1% after the 1000 trials that the coin is fair.