Irreps which aren't highest-weight modules. Well, the question is probably a rather basic one but I haven't been able to find the answer in literature or come up with it myself so here we go.
Do there exist irreducible representations of the Lie algebra $\mathfrak g$ which aren't a highest weight module with respect to some Borel subalgebra in the case when
a) $\mathfrak g$ is a complex finite dimensional semisimple algebra,
b) $\mathfrak g=\mathfrak{gl}_n(\mathbb C)$?
 A: I think it's very far from true, and misleading to state too informally, that every representation is highest weight. For example if we think of representations (with a fixed infinitesimal character) geometrically on the flag manifold (on $P^1$ for $SL_2$), then highest weight modules with respect to a unipotent subgroup $N$ are locally constant (or rather have some central curvature) along the corresponding Schubert stratification (ie for SL2 they are attached geometrically to the stratification of $P^1$ by $A^1$ and a point), while a representation can have arbitrary "shape" - ie we can fix any stratification we want. (Note the finite dimensional representations, which as was stated are all highest weight, are attached to the stratification of the flag variety by just one piece - by Borel-Weil they arise from line bundles, which definitely satisfy this "shape requirement".) In particular the representations that people care most about tend to be Harish Chandra modules, which can be realized geometrically as stratified with respect to the (finitely many) orbits of a symmetric subgroup of $G$ - for instance, the admissible representations of $SL2R$ correspond to the stratification of $P^1$ by the two poles and $C^\times$, and most of these are evidently not highest weight.
What is true, and very useful and important (some might say the most fundamental statement about g-modules, but that's not a debate I want to start), is that representations DO live on the flag manifold - in other words, they are made out of "points" of the flag manifold, which correspond to representations that are highest weight (Verma modules). So the sense in which representations relate to highest weight modules is just that they can be realized geometrically on the flag manifold ($P^1$).
Technically this is captured by a version of Casselman's submodule theorem (reps of real groups live inside principal series) due to Beilinson-Bernstein: for every representation
there IS an N so that the representation has nonvanishing N-coinvariants (ie not highest weight vectors but a form of highest weight covectors). Such N give the support of the corresponding picture of the representation on the flag manifold.
A: If you restrict yourself to finite-dimensional representations, the answer is NO.  This is discussed in most thorough Lie theory textbooks, and is a standard step in some constructions of the algebraic groups associated to the semisimple Lie algebras.  The textbook I have closest at hand is the draft Lie Groups book available on my website, based on notes from Mark Haiman's 2008 class; details are in Sections 5 and 6.  The basic idea is that the raising operators have all eigenvalues equal to zero, and so on any finite-dimensional representation must act nilpotently, and so there is a highest weight.
In infinite dimensions, the story is much richer.  In general, what people actually care about are things that can be considered representations of the group $G$ integrating $\mathfrak{g}$.  The correct notion of such representations was found by Harish-Chandra.  Details for $\mathrm{SL}(2,\mathbb R)$ are in Section 7.6 of loc. cit., among many places.  If you just want representations of the Lie algebra $\mathfrak{g}$, then in general these won't "integrate" to any representation of the group, and are a bit easier to come by.  For example, consider the case $\mathfrak{sl}(2)$ or $\mathfrak{gl}(2)$.  These act on $\mathbb C^2$ in the usual way.  On functions, we have $E = x\partial_y$, $F = y \partial_x$, and $H = x\partial_x - y\partial_y$.  Consider the module of "functions" spanned by monomials of the form $(xy^{-1})^n x^{1/3}$.  I think that this is an irrep of $\mathfrak{sl}(2)$, but I haven't checked.
A: This is covered in Is there a machinery describing all the irreducible representations ?
A: Here's a more specific reference for the rank 1 simple Lie algebra; the "classification" by Block shows clearly how sparse the highest weight representations are among all irreducible representations here.   Block was basically responding to the remarks of Dixmier and others to the effect that no "classification" would be possible, but of course his treatment is not very concrete.   
Also, the close connection between irreducible representations of a general linear Lie algebra and its derived (special linear) algebra make it obvious how close the answers to (a) and (b) are. 
A: a) No.
b) Consider determinant map.  More generally every irrep of gln is a tensor product of an irrep of sln and (integer) power of the determinant.
A: My understanding is that "formally" there are many irreps which are not highest weight (e.g. unitary principal series, Whitakker modules), however "informally" it is not true (highest weight vector is always nearby one need just to open eyes:).
The point is how do we think of representation space "V" - if we think it abstractly as some abstract vector space - okay -  can be no highest weight vector. However if we think of $V$  as some subset of functions of some variables (x_1,...,x_n), which satisfy some condition (e.g. square integrable or whatever), then we can "open eyes" and find the highest weight vector - the subtlety is that it may not obey the "condition", so it does not belong $V$, but still it corresponds to some function $f(x_1,...,x_n)$.
Example.
Take $sl_2$ and its standard representation by differential operators (sorry for some misprints):
e = d/dx
h = xd/dx - l x
f = x^2d/dx - l^2x
1) 
Consider C[x] - then it is highest weight module with "1" as an highest weight vector.
2) Consider quasi-polynoms space {f(x) = p(x)exp(m x) } 
This is actually a Whittaker module wiht character "m", and function exp(m x) is Whittaker vector i.e. it is eigen for "e".
3) Consider square integrable functions with appropriate scalar product and condition on $l$. You can get unitary principal series representation in this way.
So to my taste all these representations are essentially equivalent and they have highest weight vector "1", but "formally" it is of course not true.
