How do construction cranes rise up?

Look at rough terrain mobile crane to see if that’s what you mean.

Is this what you’re looking for?

That’s a tower crane, @thorninmud, which was not wanted.

What other kind grows up at the center of a building as it’s constructed?

We use cherry pickers (a common name for the ones I linked to) in the center of a boiler cavity all the time, until the boiler is hung low enough that it no longer fits or operates effectively. At that point the boom is retracted and lowered, the machine driven out of the building, and the boiler completed (and the leave-out steel placed where the crane drove out through one of the bays).

But those are also the only hydraulic cranes that we use.

Other hydraulic lifting mechanisms include strand jacks which are arrayed on the uppermost beams and used to incrementally (inch by inch) raise massive structures. But that would never be confused with a crane.

There are also various base-mounted jacking systems that are also “not cranes” and not to be confused with one.

The only other thing that occurs to me is that the OP has confused a crawler crane with a tower crane.

Maybe @victord66 can link us to an image of the kind of crane he’s talking about.

You can see the type of crane here: http://footage.shutterstock.com/clip-1239562-stock-footage-toronto-ontario-june-th-condo-construction-on-june-th-toronto-ontario-canada.html

The crane is actually part of the building and it moves higher as more floors are added.

That’s the kind of crane I linked to in my answer above. It’s not exactly a “part of the building” though. When the crane is no longer needed, the central tower at the core of the building will be pulled out from the top.

I agree with @thorninmud I think that’s pretty much the same crane as the ones outside the buildings.

There could be a difference in how it grows. I can’t tell if it has that hydraulic section that supports the top and pushes itself up so there is room for a new segment to be added at the top of the existing tower. Another crane might help out with that.
But at low resolution it’s hard to tell what all features it has.