I used the above code and the timer never counts above zero. I dug around some posts and saw some others had trouble because they didn't enable the clock, but I did include that line.
I thought that with the pre-scaler set that it will be counting up in microseconds so the period should be 1 and tried 1 instead of UINT16_MAX but that had no effect, count is always 0.
So after reading other posts, the documentation and trying to modify some of the setup values I still can't get this timer to count.
Does the use of systick or USART somehow disable the use of TIM5? I could not find anything that says this is true but unless there is something I missed in the code above I don't know what else to try.
The board is a STM32VL Discovery, in case that matters