User blog comment:MachineGunSuper/The Final HTN notation/@comment-30754445-20181118004650

An experienced googologist can easily see that the !'s gets you to ω3. Why? Because you have exactly three different independent "dimensions" to keep track of:

1. The "power" of the "!" you are currently using (0 for #, 1 for @, 2 for &, and in general: n for !n+1)

2. The number of times you repeat the said symbol.

3. The number that appears after these symbols.

For example to analyze "Tr&&&&7":

1. The symbol "&" corresponds to 2

2. There are 4 &'s

3. The final number is 7.

So the above function corresponds to the ordinal ω2*2+ω*4+7. And the limit of all ordinals of the form ω2*a+ω*b+c is ω3.

This notation, by the way, already beats Conway Chained Arrows (whose limit is ω2).

Now, what about your "Trω(n)"?

Well, here we have two "dimensions" that are going on simultanously:

1. The symbol you're using (!, ?, ¿, ...)

2. The number of repetitions of the said symbol.

So this portion of the notation clearly has strength of ω2. We add that to the strength of the previous section and get a grand total of:

ω3+ ω2

Looking here, you can see that this corresponds to a score of 26.6 on my Psi Levels scale (that's double the score for Graham's Number which is a bit below 13.3). So that's pretty cool.

Warning: Doubling the score again is going to be very very difficult. A score of 53.2 would require you to master the ordinal epsilon-0, which is quite a tall task.