Consider a random value which is bounded: $-a \leq X \leq a$. How large can be its standard deviation?
It's not hard to estimate this deviation from above: $$ -a \leq \mathbb E[X] \leq a \implies -2a \leq X - \mathbb E[X] \leq 2a $$ which means that standard deviation cannot be larger than $2a$. But is this bound tight? I've tried to construct such random value with deviation eqaul to $2a$, but failed.
Note that $\operatorname{Var}(X) \le \mathbb E [X^2] \le a^2$, so the standard deviation is bounded by $a$. This is sharp, since if $\mathbb P(X=a)=\mathbb P(X=-a)=\frac12$ the standard deviation of $X$ is $a$.