User blog comment:MachineGunSuper/Hypersonic Triangle Notation/@comment-30754445-20181109134629

Interesting.

The notation has the power of fω×4(n)(*) (one ω for each side on the triangle, plus another ω for the fact that we can have an arbitrary numbers on the last side used).

In an interesting twist, T△(n) is not the most powerful function in this notation. In fact, it can be shown that n△n,4n(n) is already growing much faster than T△(n). Adding many small numbers at the top of the triangle is far more powerful than putting a single huge number in there.

(*) Formally speaking, the notation is not well-defined, because the order of operations in the first two levels is ambigious (for example, a^b^a^b^...^a^b is not equal to (a^b)^^(a^b)). Fortunately, the precise definitions we choose there don't change the end result by much. Indeed, you could start with a//b=a*b and t(n)=1//2//3//...//n=n! and won't change the power of the notation (which will remain ω×4 in the FGH). The only difference is that, at the multiple-triangles level, you'll need a couple more triangles to keep up with the pace (so △100 in the new system will be about as powerful as △98 in the old one).