I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ones of $p=1$, $p=2$, and $p=\infty$.  I don't know much analysis and the best thing I could think of was Littlewood's 4/3 inequality.  In its most elementary form, this inequality states that if $A = (a_{ij})$ is an $m\times n$ matrix with real entries, and we define the norm
$$\|A\| = \sup\biggl(\left|\sum_{i=1}^m \sum_{j=1}^n a_{ij}s_it_j\right| : |s_i| \le 1, |t_j| \le 1\biggr)$$
then
$$\biggl(\sum_{i,j} |a_{ij}|^{4/3}\biggr)^{3/4} \le \sqrt{2} \|A\|.$$
Are there more convincing examples of the importance of "exotic" values of $p$?  I remember wondering about this as an undergraduate but never pursued it.  As I think about it now, it does seem a bit odd from a pedagogical point of view that none of the textbooks I've seen give any applications involving specific values of $p$.  I didn't run into Littlewood's 4/3 inequality until later in life.
[Edit: Thanks for the many responses, which exceeded my expectations!  Perhaps I should have anticipated that this question would generate a big list; at any rate, I have added the big-list tag.  My choice of which answer to accept was necessarily somewhat arbitrary; all the top responses are excellent.]

Huge chunks of the theory of nonlinear PDEs rely critically on analysis in $L^p$-spaces.


Let's take the 3D Navier-Stokes equations for example. Leray proved in 1933 existence of  a weak solution to the corresponding Cauchy problem with initial data from the space  $L^2(\mathbb  R^3)$. Unfortunately, it is still a major open problem whether the Leray weak solution is unique. But if one chooses the initial data from $L^3(\mathbb R^3)$, then Kato showed that there is  a unique strong solution to the Navier-Stokes equations (which is known to exist locally in time). $L^3$ is the "weakest" $L^p$-space of initial data which is known to give rise to unique solutions of the 3D Navier-Stokes.
In some cases the structure of the equations suggests the choice of $L^p$ as the most natural space to work in. For instance, many equations stemming from non-Newtonian fluid dynamics and image processing involve the $p$-Laplacian 
$\nabla\left(|\nabla u|^{p-2}\nabla u\right)$ with $1 < p < \infty.$ Here the $L^p$-space
and $L^p$-based Sobolev spaces provide a natural framework to study well-posedness and regularity issues.





Yet another example from harmonic analysis (which goes back to Paley and Zigmund, I think). Let
$$F(x,\omega)=\sum\limits_{n\in\mathbb Z^d} g_n(\omega)c_ne^{inx},\quad x\in \mathbb T^d,$$
where $g_n$ is a sequence of independent normalized Gaussians and $(c_n)$ is a  non-random element of $l^2(\mathbb Z^d)$. Then the function $F$ belongs almost surely to any $L^p(\mathbb T^d)$, $2\leq p <\infty$  and it does not belong almost surely to $L^{\infty}(\mathbb T^d)$.


There have been very recent applications of this resut to the existence of solutions to the nonlinear Schrodinger equations with random initial data (due to Burq, Gérard, Tzvetkov et al).
