Olympus Mons on an earth-like planet

Mars's Olympus Mons (and the three other Tharsis Montes) are very big volcanic mountains. They bulge from the surface to a ridiculous degree--Olympus Mons is three times the height of Mt. Everest, and that's on a planet that's half the size of Earth. Gravity? of course there's gravity. It would be a bit less up there, but not so much that you'd notice. Any point that's further from the planet's center of mass, including the top of Mt. Everest, has less gravity. It's just negligible. I think the confusion comes from the idea that there's zero G just a little further up, in space...which isn't true. There's gravity up there as well. The reason people on the ISS are weightless isn't because the Earth's gravity has stopped, it's because they are in free fall (they are orbiting the planet so swiftly that they are continuously "falling" around it.) You can't be in free-fall whilst standing on a mountain top--even one as tall as Olympus Mons.

Air would be very thin up there, however, as Azelor has mentioned. Only enough for the tiniest life forms, and certainly not enough for any animals like humans. That's if you're being very accurate with your planet's atmosphere.

The other thing about Olympus Mons to remember is how SHALLOW the slope of the mountain is. Seriously. You'd start walking up one side of the mountain and you wouldn't really be able to see the top. It would just be a gently rising slope in front of you. You'd keep walking, day after day, trudging over lava flows, and the air would just get thinner and thinner as you got higher up. Maybe the winds would get really nasty at certain levels, but you still wouldn't see the top of the mountain--just more slope rising in front of you. And you'd die long before you reached the top.

Which is not to say that extreme verticality is impossible. There are cliffs on Mercury that are kilometers high. It's conceivable that you could have sharp, tall, spiky mountains taller than the Himalayas on an earth-like planet. Extreme materials, extreme geology, extreme plate tectonics...it's certainly possible to have crazy-tall mountains that you *would* see the tops of as you were climbing them...as you were dying on the slopes of them...