If I have a continuous representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}_n(\mathbb{Q}_l)$ ramified at finitely many places how can the Frobenius traces behave?
Assuming they lie in an algebraic extension of $\mathbb{Q}$ can they grow exponentially with $p$? How much can $\lim\sup$ and $\lim\inf$ diverge?
"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 $\ell$-adic representations are fundamentally $\ell$-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 $\ell$. 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 (paywall, arxiv version here). The result is a list of $\ell$-adic numbers (computed modulo some high power of $\ell$), 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.
