Which is the first integer that an IEEE 754 float is incapable of representing exactly?

2^{mantissa bits + 1} + 1

The +1 in the exponent (mantissa bits + 1) is because, if the mantissa contains abcdef... the number it represents is actually 1.abcdef... × 2^e, providing an extra implicit bit of precision.

Therefore, the first integer that cannot be accurately represented and will be rounded is:
For float, 16,777,217 (2²⁴ + 1).
For double, 9,007,199,254,740,993 (2⁵³ + 1).

>>> 9007199254740993.0
9007199254740992

The largest value representable by an n bit integer is 2ⁿ-1. As noted above, a float has 24 bits of precision in the significand which would seem to imply that 2²⁴ wouldn't fit.

However.

Powers of 2 within the range of the exponent are exactly representable as 1.0×2ⁿ, so 2²⁴ can fit and consequently the first unrepresentable integer for float is 2²⁴+1. As noted above. Again.