Thinking about the things in mathematics that can describe anything that isn't entirely random, I can only think of "groups". My reasoning being, that something that isn't random must have symmetry and so the symmetry can be described by a group. Even in physics the Universe is described by groups like O(3,1) and SU(3).
Even the integers are groups under addition or multiplication.
Is there some symmetric things in mathematics that no possible group can describe?
(Having written this, I'm thinking maybe rings or vertex algebras might be a candidate?)
In order to describe the symmetries of a group $G$ we have the group $Aut(G)$ of automorphisms of $G$ which actually comes with an inner automorphism morphism $\xi: G \to Aut(G)$ which sends $g\in G$ to the inner automorphism $h \mapsto g^{-1}hg$. This structure can be written $AUT(G)$; it has the structure of crossed module which in general is a morphism $\mu: M \to P$ of groups such that $P$ operates on the right of $M$ and obeys the two rules: CM1) $\mu (m^p)= p^{-1}\mu(m)p$, and CM2) $n^{-1}mn= m^{\mu(n)}$, for all $m,n \in M, p \in P$. Crossed modules occur in discussing identities among relations for groups, and second relative homotopy groups $\delta:\pi_2(X,A,x) \to \pi_1(A,x)$. Then we can move to the automorphism structure $AUT(M \to P)$ of a crossed module!
Crossed modules also occur in the symmetry of directed graphs: see this paper by John Shrimpton.
User lysiarus also mentions groupoids, for more on which see wikipedia.