This question has already been asked twice in a similar manner (Non-Perfect Fields and Examples of fields which are not perfect). In both cases, the standard answer found in introductory textbooks was given. This is, the function fields $\mathbb{F}_q (t)$ for some finite field are not perfect.
I couldn't think of any other way to construct a non-perfect fields (except adding more variables to get $\mathbb{F}_q (t,s^q)$ or some variatons of that). I asked myself wether there are other ways to obtain non-perfect fields and in what situations these might show up.
My question thus is:
Which interesting examples of non-perfect fields do you know?
Edit. I understand that every non-perfect field must have $\mathbb{F}_p$ as it's prime field. As it cannot be algebraic, it must be transcendental. The answers and comments so far suggest that this already tells us how to get all non-perfect fields. But nonetheless: Looking at $\mathbb{Q}(i)$ we obtain a highly transcendental extension $\mathbb{C}$ in a "non-boring" way. By this I mean that we don't just add variables to get some kind of function field, but we use an analytic construction, involving some kind of metric etc. Thus, adding variables is not the only way to get transcendental extensions. This makes me wonder wether there are similar (maybe more algebraic) methods that produce transcendental extension of finite prime fields that are not perfect.
Best Answer
Fields of characteristic $0$ are perfect, and a field of characteristic $p$ contains $\mathbb{F}_p$. Algebraic extensions of $\mathbb{F}_p$ are perfect, so non-perfect extensions must contain some transcendental element $t$, i.e. they contain $\mathbb{F}_p(t)$. In this sense $\mathbb{F}_p(t)$ is the "universal" example of a non-perfect field.
Notice, however, that not every field containing $\mathbb{F}_p(t)$ must be non-perfect, just consider its perfect closure $\mathbb{F}_p(t,t^{1/p},t^{1/p^2},\dotsc)$.
If $K$ is any field of characteristic $p$, then $K(t)$ is not perfect, since $t^{1/p} \notin K(t)$.
You can also impose relations. For example, $\mathbb{F}_3(u,v)[t] / \langle t^2 + ut + v \rangle$ is not perfect.