Is there a name for this particular form of graph?

Question

Since there are so many names for particular forms of graphs, I was wondering if there is already a name for this following structure?

Suppose you construct a graph the following way:

1. Start with a complete graph on $n_1$ vertices.

2. Each of these $n_1$ vertices have $l_1$ amount of leaves (let these leaves be a set named $n_2$), where $l_1$ is at least 2.

3. Connect every vertex from $n_2$ with every other vertex in $n_2$ such that the subgraph of $n_2$ is a complete graph.

4. Each of the $n_2$ vertices have $l_2$ amount of leaves (let these new leaves be a set named $n_3$), where $l_2$ is greater than $l_1$.

5. Connect every vertex from $n_3$ with every other vertex in $n_3$ such that the subgraph of $n_3$ is a complete graph.

And so on. This can stop at any step, thus the graph can end on the attaching leaves step or at making a complete subgraph from the newly attached leaves. The only restriction that needs to hold is that $l_{i+1}$>$l_i$ from which naturally follows that $n_{i+1}$>$n_i$.

I am not a mathematician so I might have perhaps misused some notations, but I hope it is clear! Or would I have the honour to name it myself haha