Starting facts

Annihilation operator $a$ removes one photon from fock state. Coherent states are eigenstates of annihilation operator and they are also superposition of different fock states.

This leads me to following

Applying $a$ on coherent state and annihilating a photon will not change the coherent state. Therefore one can annihilate infinite number of photons from coherent state and not change state at all. This does not make much sense. Can someone point me where I am wrong?