Supplemental to copious previous discussion here: all based on this being an interesting, ie thought-provoking, question.
The crux of the problem with any asset having a uniform return distribution (as opposed to the standard assumption of normality) is that such an asset with such a distribution in one time horizon would have a very different return distribution, seen from a different time horizon.
Put simply. Over longer periods, uniform returns would then converge towards normality. But over shorter periods, returns would have to converge towards binary. And at the time horizon at which they were uniform, they would have to be bounded. So for the uniform to exist, one has to be believe in a time-dependant multiplicity of return distributions!
There need not (but might) be some arbitrage to prevent this from (theoretically) happening. However, good luck proposing the notion that your asset in question behaves "this way" daily, "that way" weekly; and "another way" monthly! This begs obvious questions about the transition dynamics between your three different asset pricing regimes. And if a clever trader can't work out how to arb that; then I'm a donkey ;-)
By claiming the uniform, you are telling me there's a point in time where the probability of price<L=0, of the price>H=0, but any point between L and H is uniformly likely. In this world, puts striked at L and calls striked at H should be free. Calls striked between these should be fractionally priced where the strike lies in the range between L and H. I can't necessarily arb you if you are 100% correct in your estimator of the return distribution... but I can get some very cheap lunches off you if you are less than 100% correct in your estimator.
Plus, I and other investors WILL leverage up, if your bounds required for your uniform hold firm. Given that we would all be simultaneous buyers/sellers at those bounds in unison, there needs to be some deus-ex-machina trader-of-last-resort willing to step in and guarantee those price levels, for the system to hold. That's obviously intuitively problematic, because it suggests that agent providing free options to market participants. If the distribution was uniform, I have NOTHING to lose buying insurance at the boundaries of your distribution; with potentially a lot to gain if the distribution is not the true one!
Over shorter time periods, one could have similar fun trading the absence of continuous behaviour. Over longer ones, trading normality versus uniformity. Put simply, the assumption of uniformity is quickly too constrained to work.