Comment by tromp

It's been shown to be surpassed at n=150, which as you note is likely very generous. Hypertree would typically only require a few more states. Hypertree doesn't grow meaningfully faster than Tree. Hypertree(3) is what would be called a Salad number, combining a very fast growing function with some very weak one(s) such as iteration, which in the Fast Growing Hierarchy corresponds to taking the successor if an ordinal.