Hmmm... I think that the functional determinants (as in http://en.wikipedia.org/wiki/Functional_determinant) of Quantum Mechanics and Quantum Field Theory appear rather naturally. These CAN be defined rigorously, insofar as the defining Feynman path integrals are, in this case, defined rigorously. The most readable introduction to rigorous path integration I've read is "A Modern Approach to Functional Integration" by John R. Klauder.

EDIT: I did some reading. The bosonic path integral expression (as in the wikipedia page I linked to earlier) for the functional determinant may fail to be multilinear (though it is rigorous), whereas the fermionic path integral expression

$\det S = \int\int \exp (\langle \bar{c} | S | c\rangle) \mathcal D c \mathcal D \bar{c}$

should be genuinely multilinear and valid for every "reasonable" S (including nonsymmetric/nonhermitian ones) IF fermionic functional integrals in infinite variables can be consistently defined to yield sensible results (e.g $\int 1 \mathcal D c=0$), which, as far as I know, has not yet been done rigorously.