(A): There exists a function f:R->R

and the axioms, for all r in R:

(A_r): f(r)=0

(The graph of f is just the x-axis.)

This might be expressible with a finite axiomatisation (e.g. by building functions and arithmetic in ZFC), and indeed I've given a finite schema, but I'm not sure it's 'fair' to ask for an example of a theory that cannot be compressed beyond uncountably many axioms; that would be a hypertask, right? I think that's what Joshua's getting at in the sibling to this comment.