What would happen if an ideal voltage source was connected to an ideal wire?
but what if both existed and they were connected together in a
  circuit?


This is entirely academic since an ideal voltage source can source or sink unlimited power for unlimited time and an ideal wire has 0V across for any current.

Thus it shouldn't be surprising that the 'answer' is not well defined:


there is 'infinite' current which allows for the possibility that an
ideal wire can have a non-zero voltage across.  In this case, the
ideal voltage source is delivering infinite power which the ideal
wire dissipates.


Clearly, this is absurd but this shouldn't be surprising since, in fact, the equation this ideal circuit yields is (assuming a 1V voltage source)

$$1V = 0\Omega \cdot I_S $$

which has no solution for any finite \$I_S\$ which is, in fact, the most correct answer in my opinion.

The fact is, there are rules for connecting ideal circuit elements to avoid such contradictory mathematical expressions including the rule that one does not connect two ideal voltage sources in parallel.

Yes, that rule covers this case since an ideal wire is identical to an ideal voltage source with voltage equal to zero.

Surprised?  But that's how an ideal voltage source is defined - an ideal voltage source is that two terminal circuit element with a given voltage across for any current through.

Set the given voltage to zero and you have an ideal wire.  In fact, this is why, in superposition, when voltage sources are zeroed they are replaced by wires.