> "Assuming they're in an algebraic extension of $\mathbb{Q}$ can they
> grow exponentially with 𝑝?"?

I'm not sure this statement will have a truth value, because I suspect the assumption never occurs: non-geometric representations are fundamentally p-adic analytic objects and I don't know of any mechanism which would force their Frobenius traces to be in $\overline{\mathbb{Q}}$. 

[Here I'm assuming that your representations are irreducible; one can easily construct examples of extensions $0 \to \chi_1 \to V \to \chi_2 \to 0$ with $\chi_i$ geometric which are unram almost everywhere but non-geometric at $p$. But then the traces are the same as $\chi_1 \oplus \chi_2$ so the question is not interesting.]

There are examples of irreducible 2-dim'l non-geometric representations arising from non-classical overconvergent modular forms. I tabulated a bunch of these in [my first ever paper][1] (paywall, arxiv version [here][2]). The result is a list of p-adic numbers (computed modulo some high power of p), which are the q-expansion coefficients, or equivalently Frob traces at the first few primes; and they don't like they come from $\mathbb{Q}$ in any recognisable way.


  [1]: https://academic.oup.com/imrn/article-abstract/doi/10.1093/imrn/rnm050/673208?redirectedFrom=fulltext
  [2]: https://arxiv.org/abs/math/0701168