Assuming there are exactly $365$ days per year. How many rotations about its axis does the earth make in one year?
As the Earth revolves around the Sun it has to turn slightly more than a complete rotation each day to account for the extra angular distance that it has traveled. Over the course of a year, this will add up to an extra sidereal day over the number of solar days.
Half jokey/ half serious answer:
Exactly one rotation. Assuming the axis in question is the one through the gravitational center of the Earth/Sun system. One rotation around that axis is the definition of a year.
Something like .000000004 rotations around the Milky Way galactic center.
Mama never told be not to play with axes