See http://math.uni.lu/~wiese/galois/BoeckleLuxemburgNotes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$representation $V$ of $G_F$ ($F$ a global field) ramified at a set $S$ of places where $S$ is not finite (for every $\dim{V} \geq 1$)?
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In the case $\dim V=2$, my advisor Ravi Ramakrishna has shown in his paper Infinitely ramified Galois representations (Ann. Math. 151) that there are surjective representations $\rho\colon G_\mathbf{Q}\to \mathrm{GL}_2(\mathbf{Z}_p)$ ramified at infinitely many primes. 


It seems that for $\dim{V} = 1$, there is no such representation http://www.math.leidenuniv.nl/scripties/KretMaster.pdf p. 10. 

