Do more objects appear on sky in time? I would assume that, with light from more distant areas in space reaching the earth, we should be able to see more distant (and older) objects.
However there's lot of weird stuff regarding the space expansion that could eventually make the objects leaving us so fast the we'd never see them.

So, did we reach any boundary of what we can possibly see, or are we still working on more sensitive exploration devices?

I'll be grateful for some useful re-tag
No, the expansion of the universe appears to be accelerating. Hence distant objects get more and more red-shifted. Our horizon is shrinking instead of expanding. At the end, only galaxies which are sufficiently bound by gravity remain visible; this could turn out to be the Virgo supercluster.

More reading about the accelerating universe on Wikipedia, and on National Geographic.

Scientific background to the according Nobel Prize in physics 2011.