To the best of my knowledge, no, but maybe other people here know the field more intimately. My knowledge comes primarily from developing Monte Carlo codes in physics.
Knuth, in volume 2 of his Art of Computer Programming, states that Metropolis, using the middle-square method on 20 bits, found 13 cycles to which the method would always degenerate, the longest of which has a period of 142. Sadly I've been unable to find the original paper, but note that he only refers to the period of the cycle, not the time it takes to reach the cycle from a given period.
More generally, most (but not all) random number generators used in numerics today have a guaranteed period, often independent of the seed. This period is however calculated differently from generator to generator.
See this for a list of reasonable random number generators.
Finding the period experimentally will be very difficult for almost all modern random number generators, simply because the periods are very very long. In many cases it would take far far longer than the expected age of the universe, even if every single atom was turned into a computer to work on the problem in parallel. Just to give you an idea of how difficult it is.
Anyway, for toy random generators, remember that in order to find a cycle, you need to check if the entire state space of the generator is repeating. In the case of the middle-square method, the state and the output are the same, but this is not the case for most random number generators, and certainly not for any practical ones.
PS: The middle-square method is a really bad random number generator and should not be used for generating random numbers. Ever.