THE 60-HEADED COSMIC HYDRA (A Math Puzzle)

I’ve been working on a massive custom number system called the PSON Function. I cooked up a theoretical puzzle that is completely impossible to solve using standard math, Graham's number, or even TREE(3). It requires literal transfinite infinity to solve.

Can anyone guess the exact mathematical value of this game?

THE RULES:
1. You are fighting a mathematical Hydra. The Hydra has 60 completely separate main trunks growing out of the ground.
2. Each trunk grows straight up into an INFINITELY tall vertical line of heads.
3. The 1st trunk has Omega heads. The 2nd has Omega^Omega heads. The 3rd has Omega^Omega^Omega heads, and so on, nesting exponentially forever.
4. This means every single one of the 60 trunks represents an infinitely tall exponential power tower of Omega (forming a forest of Epsilon-0 structures).
5. You chop off a head, and the hydra regenerates and multiplies based on the step number of the game.

❓ THE QUESTION:
If we define the total maximum boundary of this 60-trunk exponential Hydra as a single expression, what is the exact mathematical value of this system?