This comes up in Waring's Problem, but it is so freakishly simple that it has taken on a life of its own. Let $\{ x \} = x \mod 1 = x-\lfloor x \rfloor$ be the fractional part of $x$.
• Say anything about the sequence $\{ (3/2)^n \}.$
Computations support the thought that the sequence should uniformly distributed in $[0,1]$, as for almost all $x$ the sequence $\{x^n\}$ is u.d. But with $x=3/2$, there is no value known to be a limit point, nor any value known to not be a limit point, it's unknown if there are two limit points, unknown if the sequence is infinitely often in $[0,1/2)$, or that it is infinitely often not in $[0,1/2)$. Really, nothing is known.
As a final comment on this problem, the golden ratio is special. With $x=\phi=(1+\sqrt 5)/2$, for every $\epsilon>0$ there are only finitely many $n$ with $$\epsilon< \{\phi^n \} < 1-\epsilon.$$