Does a planet's mass or gravity affect the height of it's mountains? According to this Wikipedia page, the top five highest mountains on Mars (and the highest on Venus) are all taller than Mount Everest (and Mauna Kea as measured from the ocean floor).

Does a planet's mass or gravity affect the height of it's mountains?  Is there an upper limit that a mountain can reach given the mass or gravity of a planet?
Yes, gravity definitely affects the maximum heigh of mountains.

Think on a solid bar of steel. It sticks straight because of the electronic forces. But when you make it larger and larger gravity makes it bend: gravity starts being considerable, but still smaller than electronic forces.

If you make the bar larger, there will be a moment in which the weight of the whole bar will be larger than the short-range electronic force: your bar will break purely due to gravity.

Exactly the same happens to mountains made of solid rock (as opposed to sedimentary ones quoted by Hobbes). There is a point,k depending on the strenght of planetary gravity, where it takes over short-range electronic forces, making the mountain collapse.

This is exactly the force that "rounds up" the planets, as opposed to the non-spherical asteroids.