This depends on the sense in which you mean the word 'useful' -- mathematicians try to make a study of the logical objects which are 'useful' or interesting in some sense, and pretty much any set-theoretic encoding (or type-theoretic encoding) of this information will require assumptions, which are identical to axioms from the standpoint of philosophy of set theory.
Perhaps this is not true by necessity, though. There is currently a development underway that is referred to as the 'reverse mathematics of second-order set theory', which seeks to understand what assumptions are necessary to prove certain well-known theorems in mathematics. This is a reversal of classical mathematics, in the sense that we are moving from a place of
'we believe this theorem is true in most reasonable contexts -- what is the bare minimum that we need to assume to prove it so',
as opposed to
'here is what we are assuming is true, what are the consequences'.
A development with minimal axiomatic use for maximum proof capability would probably fit somewhere into this hierarchy of assumption strength. I don't know how one would proceed with no assumptions, however -- even your suggestion of Occam's razor is ultimately still just an assumption.