Which astronomer first measured the distance to another galaxy?

Edwin Hubble, in 1924, and the measurement ended the biggest argument in astronomy. Nobody knew whether the spiral nebulae were clouds inside our own galaxy or separate galaxies far away, and it had been formally debated in 1920 with respectable astronomers on both sides.

He settled it with a particular kind of variable star. Cepheids pulse with a period that depends on their true brightness - a relationship discovered by Henrietta Swan Leavitt, who worked as one of the Harvard computers, women employed to measure photographic plates at low wages and largely denied authorship. Knowing the true brightness and measuring the apparent brightness gives the distance. Hubble found Cepheids in Andromeda and calculated it was far outside the Milky Way.

The universe got enormously larger in one measurement, and then Hubble did it again - showing that galaxies are receding at speeds proportional to their distance, which is the expansion of the universe. He never won a Nobel Prize because astronomy was not then eligible under the physics category, and he lobbied for years to have that changed. It was changed shortly after he died.