Turing machine with a countably infinite alphabet
Your model accepts all possible languages over the countably infinite alphabet.

The first step is to convert the input word to a single symbol, in some one-to-one fashion. We can do that since $\bigcup_{n=0}^\infty \aleph_0^n = \aleph_0$. You can then encode your language in the transition function.

More generally, to get a non-trivial model you need to ensure that the description length of the machine is smaller in cardinality than the number of words over the alphabet.