It seems the question is not so idle after all, for the Fontaine-Mazur conjecture has something to say about it, as I discovered today in Neukirch-Schmidt-Wingberg (the NSW of Cam's answer).  To paraphrase their (10.8.13), *every representation*
 
$$
\operatorname{Gal}(M|K)\to
\operatorname{GL}_n(\mathbf{Q}_p)
$$

*of the group of automorphisms of the maximal unramified extension* $M$ *of a number field* $K$ *has finite image*, even if the degree $[M:K]$ is infinite.  This is certainly a very strong restriction.