Considering how old the Antarctic ice cover is, why isn't it much thicker? The Wikipedia article on the subject of the "Antarctic ice sheet" says that:


  The icing of Antarctica began in the middle Eocene about 45.5 million
  years ago and escalated during the Eocene–Oligocene extinction event
  about 34 million years ago.


The article later says that:


  Ice enters the sheet through precipitation as snow. This snow is then
  compacted to form glacier ice which moves under gravity towards the
  coast.


However it looks like, from the photos of Antarctica, that this transfer of ice to the coasts is not happening everywhere on Antarctica. Rather, many places seem to be under a perpetual ancient ice cover.

If the precipitation was recycled at a continuous rate everywhere, it shouldn't be possible to obtain a 1.5 million year old core sample (according to an article in Nature) or structures such as Vostok Station that covers an ancient lake.

A similar location is Dome F that according to Wikipedia has a yearly precipitation of about 25 mm (millimeters; approx. 0.98 inches).

So the question is, if we have a 25 mm precipitation per year for even just 10 million years, we should have an ice cover that's 250 kilometers thick. Or if we had it during just a 5 million year time span, it should give us an ice cover that's 125 kilometers thick.

Reasonably we could allow for some compression of the ice but still keeping in mind that we are not talking about a neutron star here.

How come the current Antarctic ice cover is just a couple of miles thick?
Ice floats with gravity towards lower elevation, the flow continues until the base of the ice sheet becomes floating and the ice forms an ice shelf or icebergs. 

Due to the subglacial topography, basal melting and mass balance, the flow velocities vary over a large range, faster outflows are glaciers. The pattern is somehow similar to how rivers transport rainwater towards the coasts. 

The thickness of the ice sheet is controlled by a complex relation between accumulation, ablation and compaction.  


Flow velocities from MEaSUREs (NASA), plotted from Quantarctica

The ice velocity map shows, that some areas are unaffected by the flow towards the coast. Some of these areas are ice-free. Also at some inland locations, as Vostok, the velocity is very slow. Also, note that this is the surface velocity. At the base, the ice flows slower or not at all. 

To investigate the mass balance further, I recommended you to download Quantarctica and look closer at the datasets. Shepherd et al (2012) is a good first read about the methods used to estimate changes in ice mass and the present changes.