4 x 3 = 4 + 4 + 4
4 ^ 3 = 4 x 4 x 4
4 ^^ 3 = 4 ^ 4 ^ 4
4 ^^^ 3 = 4 ^^ 4 ^^ 4
4 ^^^^ 3 = 4 ^^^ 4 ^^^ 4

They all follow the same pattern. For up arrows, you just have one less up arrow than in the original problem.

You are correct that

3 ^^ 4 = 3 ^ 3 ^ 3 ^ 3

However,

3 ^^^^ 3
is actually = 3 ^^^ (3 ^^^ 3)

Working on the parentheses first, you'd find that (3 ^^^ 3) = 3 ^^ (3 ^^ 3)

Again, working on the parentheses, (3 ^^ 3) = 3 ^ 3 ^ 3 = ~7.6 trillion. Put that back in your other problem

and you've got 3 ^^ (~ 7.6 trillion), which means you're doing 3 ^ 3 ^ 3 ^ 3... until you've done it 7.6 trillion times.

Once you get that crazy huge number, you'd put it back in the original problem:

3 ^^^ (crazy huge number).

Which is 3 ^^ 3 ^^ 3 ^^ 3... until you've done it a crazy huge number of times. Unimaginably big. And that's only with only 4 up arrows.