That's hard to tell, because already the next one after tetramethylbutane, tetra-tert-butylmethane (3,3-di-tert-butyl-2,2,4,4-tetramethylpentane) is so unstable it cannot exist. You want a parent chain that is even two atoms longer.
Actually it seems also removing one methyl group from $\ce{tBu_4}$methane doesn't make it sufficiently stable, so one has to be a bit more generous in weeding out branches for larger hyperbranched alkanes.
I believe that any permethylated (or per-n-alkylated) linear chain is principally possible, but adding more side-side chains to adjacent side chains will make the molecule snap in two.
So the biggest possible sum formula would be something based on
3,5-di-tert-butyl-2,2,3,4,4,5,6,6-octaamethylheptane = $\ce{C23H48}$
. You can surely make the inner methyl substituents longer,
3,5-di-tert-butyl-3,5-diethyl-4,4-di-n-propyl-2,2,6,6-tetraamethylheptane = $\ce{C29H60}$
, and then you can probably add 4 methyl to the nPr and Et to give iBu and i-Pr that's $\ce{C33H68}$. Any more and I'll bet it breaks, quite a way from the theoretical $\ce{C53H108}$.