I've certainly never found a common UI pattern that isn't describable with FSMs, but you could probably think of a few toy examples of non-terminating state. Anything involving recursion, exponential, or asymptotic behavior for example could theoretically involve infinite states. Again, likely of no practical use.
Read any real-world user interface specification like Apple's classic Macintosh Human Interface Guidelines [1]. It specifies all kinds of complex behaviors, like how pull-down submenus don't pop up immediately as you move the cursor over them, and how dragging and dropping text handles the white space at the beginning and end of the selection, or how adjusting the end of the selection in mixed right-to-left and left-to-right text works, which is much too complex and nuanced and ad-hoc hacky to describe with finite state machines, and require a Turing complete machine to implement.
Even if it were technically possible to restrict yourself to a FSM, you would end up going insane and producing exponentially explosive unmaintainable finite state spaghetti machines.
There's a huge uncanny gap between what you can elegantly describe with a clean mathematical abstraction like finite state machines, and what real human beings expect and perceive to be easy to use in a real world user interface.
The polish of a good user interface is the result of millions of tiny little scratches, and in practice that requires the power and flexibility of a Turing machine (and not one stuck in the Turing tarpit like Brainfuck) to make all of those tiny little scratches and cover all the edge cases and special circumstances, because finite state machines are just too clumsy and not powerful enough to conveniently cover that much ad-hoc messy fractal detail.
