[darkmighty]

There are also numbers that can't _possibly_ be represented by this system using the elementary functions/operators, i.e. are "non-analytic".

This may actually not be a problem in most instances, however. It really depends on the use case, but outside of scientific computing/engineering design we're not usually doing computations to find roots of degree >= 5 polynomials.

OTOH, as pointed out by GP, it is easy to construct 'unsimplifiable' operations which after some time may grow too large.

My conclusions are, this certainly cannot be implemented on practical systems without some care -- it's not as straightforward as it looks at first -- and certainly not a better default as is right now.