How many planes can be passed through one point?

Hypothetically, an infinite number of planes.

We humans develop contrived measurements, such as 360 degrees in a circle, but that does not mean there are only 360 possible angles; even subdividing as we do with 60 minutes in a degree, and 60 seconds in a minute, there is no hypothetical smallest fraction of an angle. You could have one-billionth of a degree of an angle to differentiate one plane from another passing through the same point, or one-quadrillionth, or one-googoleth, or one googole-plexeth, ad-infinitum (-eth).