How are the hyperoperations defined? For the first few operations such as addition and multiplication they follow rules such as
$$A(x,y+1)=A(x,y)+1$$
$$A(x,0)=x$$
For multiplication
$$M(x,y+1)=M(x,y)+x$$
$$M(x,1)=x$$
So for a general hyper operation you would think that
$$\Pi_n(x,y+1)=\Pi_{n-1}(\Pi_n(x,y),x)$$
But then this doesnt work for tetration because
$$\Pi_4(x,y+1)=\Pi_{3}(\Pi_4(x,y),x)$$
$$^{y+1}x =(^{y}x)^x$$
But this implies
$$^{y}x =(x)^{x^y}$$
Because exponents multiply
When it should be
$$^{y}x =x^{x^{x^{x^{x^{{...}}}}}}$$
What should be the correct formula for the nth hyperoperation.
Thanks.
 A: 
What should be the correct formula for the nth hyperoperation?

Well, there is no answer because both of them are currently called Hyperoperations.
Your definition gives what are called lower-hyperoperations.

The recursive relation for lower h.os is the one you give:
$$h_{n+1}(x,y+1)=h_n(h_{n+1}(x,y),x)$$
As you can read on Wikipedia, this is only one of the possible definitions and probably not the most natural one.
The "natural hyperoperations", as I call them, or Goodstein's hyperoperations satisfy instead this relation:
$$H_{n+1}(x,y+1)=H_n(x,H_{n+1}(x,y))$$
Goodstein himself, who proposed the term tetration, pentation , ecc, defines them in this way in R. L. Goodstein (Dec 1947). "Transfinite Ordinals in Recursive Number Theory". Journal of Symbolic Logic. 12 (4): 123–129.

It's easy to prove that both definitions agree on the first 3 operations, and diverges at the fourth.

Is this the correct definition? We can't know unless we specify what
we mean with correct. It's not the oldest definition. For sure it's
the most natural and, I claim, the most fundamental one.
A: Just a comment, not an answer.
I've once given it a try, but gave it also an operator-notation (=adapted to look more like the binary $+$,$*$ - notation). Also looked at neutral and annihilating elements. But didn't fill all table-entries. Didn't took the final conclusion what pleased me most... See this

