Biggest Field Of Characteristic $p$
Conway's nimbers form an interesting answer for $p=2$. That every Field of characteristic $2$ embeds into it follows from the fact they form an algebraically closed Field and that they contain arbitrarily large sets of algebraically independent elements (which is immediate because the Field is proper-class-sized).
This has been generalized by DiMuro to arbitrary $p>0$ in his paper "On $\mathrm{On}_p$", see here. The resulting Field again is algebraically closed, and every characteristic $p$ Field embeds into it for the same reasons as above. The construction is more artificial than that of nimbers, but still.
By the way, let me make a remark: all the Fields mentioned in the thread, that is surreals, surcomplexes, nimbers and $\mathrm{On}_p$, are not unique maximal fields, even up to isomorphism. Indeed, for instance, taking the field of rational functions $\mathrm{On}_p(x)$, it is clearly not algebraically closed, hence not isomorphic to $\mathrm{On}_p$, yet every characteristic $p$ Field will embed into it (since they embed into $\mathrm{On}_p$). This does not change the answer, but I thought it's worth emphasizing.
To address the newly added question and the comment below, I do not know of any sense in which $\mathrm{On}_p$ is minimal - it contains proper subFields which also have the property that other Fields embed into it. However, as the other answer says, this Field is unique (up to isomorphism) characteristic $p$ Field with this property which is algebraically closed (indeed, this is a unique algebraically closed Field in characteristic $p$). This also is true of surcomplexes in characteristic $0$. I don't know of a corresponding description for surreals - there are many real closed fields of given cardinality, but see Alec Rhea's comment.
An algebraically closed field is determined up to isomorphism by its characteristic and its transcendence degree over its prime field. So every algebraically closed field of characteristic $p$ is isomorphic to the algebraic closure of $\mathbb{F}_p(X)$, where $X$ is some set of variables.
This suggests that the "biggest field of characteristic $p$" should be constructed in the same way, but with a proper class of variables. e.g. the algebraic closure of the field of rational funtions over $\mathbb{F}_p$ in variables $(x_\alpha)_{\alpha\in \text{Ord}}$.
Under global choice, this is the unique proper class sized algebraically closed field of characteristic $p$ up to isomorphism.
It's up to you whether you view this as a "nice characterization".
As an aside: it's quite common in model theory to consider a proper class sized "monster model" for any complete first-order theory $T$. The monster model has the property that every set sized model of $T$ embeds into it elementarily. So the construction of monster models answers the analogue of your question for any complete first-order theory $T$.