TITLE: Biggest Field Of Characteristic $p$
QUESTION [16 upvotes]: The Surreal nummbers, $\boldsymbol{No}$, are according to Wikipidia the biggest ordered field, and the Surrcomplex numbers are the biggest field of characteristic 0. Biggest in the sense that every field (which is a set) is imbedded inside. 
In light of that I was wondering whether there is a nice characterization of the biggest field of characteristic $p > 0$?
As Wojowu noted, this isn't unique to the Surreals or to $On_p$. The question then becomes is there a minimal such field?

REPLY [19 votes]: 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 $T$.