Uncountable Set of Poles?
This really comes down to how you define LTI.
Far as I can tell, a time-discrete system being LTI means its in- and output are related by a countably large system of ordinary differential equations.
Since each of these transforms into a polynomial through the Laplace-, Fourier- or z-transform, the transfer function, in which ever domain it exists, is always a quotient of polynomials. Coefficients of polynomials are countable.
Since polynomials have, including their multiplicity, exactly the number of roots as their degree, and poles are just the roots of the denominator polynomial, we'll always have countably many roots – and thus, poles.

Not quite sure you'd want to relax the constraint that poles be countable, anyway: If you do that, transfer functions cease to be meromorphic, and that means the residue theorem doesn't apply, and as far as I can intuit, that leads to unsolvable integrals that you'd need to define the $z$-transform to begin with...