On STM32F4xx: Can I trigger a timer by itself so it goes through a list of periods updated from an array?
I'm not sure what exactly you mean by "triggering itself", do you mean up- or down-counting, from a fixed clock?

I have not done this myself, but if I'm understanding your question correctly then you're looking for the auto-reload register (ARR). The counter can be configured to be reset once it reaches the ARR value, and this also sets an event flag (UDE or UIE bit in TIMx_DIER). You can service this flag either in an ISR or with a DMA channel, each time updating ARR from a table of periods. "Stopping" can be achieved with ARR = 0.

Also see ST's appnote AN4013 about the STM32 timers, which includes example setups for some common use cases.