I'm studying an infinite (nonabelian) group $G$ which can be written as an infinite nested union; $$ G = \bigcup_{j=1}^\infty G_j, \quad \{ 1 \} \subset G_1 \subset G_2 \subset \cdots $$ with the property that for any $g \in G_j$ and $h \in G_k$ we have $h \ast g \in G_{j+k}$. I have a vague memory of seeing such groups studied in some book or paper, but googling "stratified groups" etc has beared no fruit.
Is there a standard name for such groups?
On
I would call this a "filtered group," by analogy with filtered algebras.
An easy way to produce such a filtration is to pick a set of generators $S$ of $G$ and define $G_i$ to be the elements of $G$ which can be generated by a product of at most $i$ of the generators in $S$, or equivalently the elements of length at most $i$.
I think you might be thinking of a directed union of groups, although it satisfied a stronger property than what you gave, namely that $G_iG_j\subseteq G_{max(i,j)}$. You may also consider the groups involved in the union a filtration of the group $G$.
A directed union is a special case of a direct limit where you know the limit is just going to be the union of the elements of the base of the cone, and the morphisms between elements of the base of the cone are all inclusion maps.
Perhaps also you were thinking of something along the lines of a graded ring. Graded algebraic structures tend to have the closure axiom you give (which I somehow missed when I originally wrote my answer. I thought you just had the union.)
Added: I also did not realize that filtered structures had the operation compatibility as well. I thought the chain was the only thing that mattered. So it looks like my ideas about grading and filtration cover the same ground.