The FontaineMazur conjecture predicts that an $l$adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over $l$. Now, what happens if one forgets about the latter restriction; are there any counterexamples, and is there any (general?) way to understand that those are not geometric without using the De Rham restriction?

To complete Kevin's good answer: the number of $\ell$adic representations (up to isomorphism) of a number field $K$ is countable, since so are varieties over a $K$. On the other hand, we know by Mazur's theory of deformations that representations of the type you consider that is, of the Galois group of the maximal extensions of a number field $K$ unramified outside a finite set of places $S$ containing places above $\ell$ and $\infty$) form multidimensional $\ell$adic family (e.g. parametrized by spaces like $\mathbb{Z}_\ell^n$ for some $n>0$, hence are uncountable. Thus not only are there counterexamples to FontaineMazur's conjecture without the de Rham hypothesis, but most examples of such representations are counterexamples. Among all representations, the de Rham (or geometric) representations are expected to be dense in certain cases (e.g. representation of dimension 2 of $\mathbb Q$) but not in general (e.g. representations of dimension $2$ of a nontotally real number field) 


In fact there are onedimensional counterexamples: if $\chi$ is the $l$adic cyclotomic character, and $k\in \mathbb{Z}_l \backslash \mathbb{Z}$, then $\chi^{(l1)k}$ is unramified outside $l$, but does not arise from geometry (in particular it is not de Rham). A general way of understanding 'all' Galois representations is via Mazur's universal deformation rings. These parametrize all $l$adic Galois representations (unramified outside a prescribed set of primes and perhaps with extra conditions) whose reduction mod $l$ is isomorphic to a given mod $l$ representation. The $R=T$ theorems you may have heard of state that certain universal deformation rings are isomorphic to rings coming from the theory of modular forms. I don't know if it's expected in general whether all Galois representations are limits of deRham representations. 

