Perhaps the simplest wavelet I've found, in my cursory glance over the field, is the Haar wavelet: $$\psi (t) = \operatorname{sgn}\left(t - \frac{1}{2}\right)\, \Theta(t)\, \Theta(1 - t),$$ with $\operatorname{sgn}$ the sign function, and $\Theta$ the Heaviside step function.
That function seems more complicated than a the simpler boxcar one, though: $$\psi (t) = \Theta\left(t + \frac{1}{2}\right)\, \Theta\left(\frac{1}{2} - t\right).$$
Is this considered a wavelet? If so, what's it's name? If not, why not? Part of why I ask is because Wikipedia describes the Shannon wavelet as the Fourier dual of the Haar wavelet, when it seems like the boxcar wavelet is more accurately described as its dual.