Uncountable Set of Poles? It is easy to define an (ideal) LTI system that would have an infinite number of poles - for instance, if the transfer function is
$$
H(z)=\frac{1}{\cos(z)-1}
$$
However, this would only define a countably infinite set of poles.
I am curious: are there systems with uncountably infinite sets of poles?
I would guess that the very definition of a pole would prohibit a function from having a contiguous region of poles, but what about a hypothetical transfer function that would have its poles defined on something like Cantor set?
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...