The obvious ones are 0 and $e^{x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?
The following is discussed in a little more detail on pages 337339 of Frank Jones's book "Lebesgue Integration on Euclidean Space" (and many other places as well). Normalize the Fourier transform so that it is a unitary operator $T$ on $L^2(\mathbb{R})$. One can then check that $T^4=1$. The eigenvalues are thus $1$, $i$, $1$, and $i$. For $a$ one of these eigenvalues, denote by $M_a$ the corresponding eigenspace. It turns out then that $L^2(\mathbb{R})$ is the direct sum of these $4$ eigenspaces! In fact, this is easy linear algebra. Consider $f \in L^2(\mathbb{R})$. We want to find $f_a \in M_a$ for each of the eigenvalues such that $f = f_1 + f_{1} + f_{i} + f_{i}$. Using the fact that $T^4 = 1$, we obtain the following 4 equations in 4 unknowns: $f = f_1 + f_{1} + f_{i} + f_{i}$ $T(f) = f_1  f_{1} +i f_{i} i f_{i}$ $T^2(f) = f_1 + f_{1}  f_{i}  f_{i}$ $T^3(f) = f_1  f_{1} i f_{i} +i f_{i}$ Solving these four equations yields the corresponding projection operators. As an example, for $f \in L^2(\mathbb{R})$, we get that $\frac{1}{4}(f + T(f) + T^2(f) + T^3(f))$ is a fixed point for $T$. 


Following on a little from Andy's comment, Hermite polynomials (multiplied by a Gaussian factor) give a basis of eigenvectors for the FT as an operator on $L^2({\mathbb R})$ 


A very important fixed point of the Fourier transform that isn't in $L^2$ is the Dirac comb distribution, informally $$D(x) = \sum_{n\in Z} \delta(xn),$$ or more properly, defined by its pairing on smooth functions of sufficient decay by $$\langle D, f\rangle = \sum_{n\in Z} f(n).$$ The fact that $D$ is equal to its Fourier transform is really just the Poisson summation formula. (I wrote an argument explaining why $D$ should be its own Fourier transform in an answer to another question: Truth of the Poisson summation formula) 


A more complete list of particular selfreciprocal Fourier functions, i.e. eigenfunctions of the cosine Fourier transform: $1.$ $\displaystyle \frac{1}{\sqrt{x}}$ $2.$ $\displaystyle e^{x^2/2}$ (and more generally $e^{x^2/2}H_{2n}(x)$, where $H_n$ is Hermite polynomial) $3.$ $\displaystyle\frac{1}{\cosh\sqrt{\frac{\pi}{2}}x}$ $4.$ $\displaystyle \frac{\cosh \frac{\sqrt{\pi}x}{2}}{\cosh \sqrt{\pi}x}$ $5.$ $\displaystyle\frac{1}{1+2\cosh \left(\sqrt{\frac{2\pi}{3}}x\right)}$ $6.$ $\displaystyle \frac{\cosh\frac{\sqrt{3\pi}x}{2}}{2\cosh \left( 2\sqrt{\frac{\pi}{3}} x\right)1}$ $7.$ $\displaystyle \frac{\cosh\left(\sqrt{\frac{3\pi}{2}}x\right)}{\cosh (\sqrt{2\pi}x)\cos(\sqrt{3}\pi)}$ $8.$ $\displaystyle \cos\left(\frac{x^2}{2}\frac{\pi}{8}\right) $ $9.$ $\displaystyle\frac{\cos \frac{x^2}{2}+\sin \frac{x^2}{2}}{\cosh\sqrt{\frac{\pi}{2}}x}$ $10.$ $\displaystyle \sqrt{x}J_{\frac{1}{4}}\left(\frac{x^2}{2}\right)$ $11.$ $\displaystyle \frac{\sqrt[4]{a}\ K_{\frac{1}{4}}\left(a\sqrt{x^2+a^2}\right)}{(x^2+a^2)^{\frac{1}{8}}}$ $12.$ $\displaystyle \frac{x e^{\beta\sqrt{x^2+\beta^2}}}{\sqrt{x^2+\beta^2}\sqrt{\sqrt{x^2+\beta^2}\beta}}$ Examples $15,810$ are from the chapter about selfreciprocal functions in Titschmarsh's book "Introduction to the theory of Fourier transform". Examples $11$ and $12$ can be found in Gradsteyn and Ryzhik. Examples $6$ and $7$ are from this question What are all functions of the form $\frac{\cosh(\alpha x)}{\cosh x+c}$ selfreciprocal under Fourier transform? that I started at Math stack exchange. I'm trying to collect all such interesting examples in one place, so if someone knows other examples, please feel free to add them in this post or in comments below. 

