The problem can be equivalently (and even more canonically) posed with argument 0

a [0] 0 = 1, no right neutral element e, x[0]e=x => e=x-1

a [1] 0 = a, neutral element 0

a [2] 0 = 0, neutral element 1

a [3] 0 = 1, right neutral element 1, no left neutral element e[3]x=x => e=x^(1/x)

a [4] 0 = 1

a [3L] 0 = a

And now we also see the pattern. For each operation we choose as initial value the neutral element of the corresponding side of the sub/preceding operation, if there is any. Otherwise we choose a, the left operand, as initial value. (*)

"the corresponding side" means the left side for the left-bracketed/lower hyperoperation and means the right side for the right-bracketed/upper/normal hyperoperation.

To look at some deviant hyper operations, let us define (only for this post, hopefully we elaborate a more refined notation later):

a[nr]0=right neutral elment of [n]

a[nR]0=a

a[nl]0=left neutral element of [n]

a[nL]0=a

If we now start with the initial operation a[0]x=x+1, we have the following hierarchy:

a[1]x=a[0R]x=a+x

a[2]x=a[1r]x=ax

a[3]x=a[2r]x=a^x

a[3L]x=a[3L]x=a^a^x

and the following deviant operations:

a[1R]0=a, a[1R]1=a[1]a=2a, ...

a[1R]n=a(n+1)

There is no left neutral element e, e[1R]x=x => e=x/(x+1).

0 is the right neutral element

a[1Rr]0=0, a[1Rr]1=a[1R]0=a, a[1Rr]2=a[1R]a=a(a+1)=a^2+a, a[1Rr]3=a[1R](a^2+a)=a^3+a^2+a, ...

a[1Rr]n=

There is no left neutral element e, e[1Rr]x=x.

1 is the right neutral element

How can this be extended to the real n?

Deviant tetration

a[1Rrr]0=1, a[1Rrr]1=a[1Rr]1=a, a[1Rrr]2=a[1Rr]a=?

a[2R]0=a

a[2R]1=a a=a^2

a[2R]n=a^(n+1)

There is no left neutral element

0 is the right neutral element

Another deviant tetration:

a[2Rr]0=0, a[2Rr]1=a[2R]0=a, a[2Rr]2=a[2R]a=a^(a+1), a[2Rr]3=a^(1+a^(a+1)), ...

We saw already that the sub operations below [0] are all equal to [0], i.e. increments. And those are unfortunately the operation(s) that dont fit into this scheme. Because the initial value to go from a[-1]x=x+1 to a[0]x=x+1, is a[0]0=1, though neither 1 is a right neutral element of [-1] nor is 1=a.

But this can be reformulated that there are no operations below [0] that obey our general rule (*), i.e. the hierarchy starts at 0.

a [0] 0 = 1, no right neutral element e, x[0]e=x => e=x-1

a [1] 0 = a, neutral element 0

a [2] 0 = 0, neutral element 1

a [3] 0 = 1, right neutral element 1, no left neutral element e[3]x=x => e=x^(1/x)

a [4] 0 = 1

a [3L] 0 = a

And now we also see the pattern. For each operation we choose as initial value the neutral element of the corresponding side of the sub/preceding operation, if there is any. Otherwise we choose a, the left operand, as initial value. (*)

"the corresponding side" means the left side for the left-bracketed/lower hyperoperation and means the right side for the right-bracketed/upper/normal hyperoperation.

To look at some deviant hyper operations, let us define (only for this post, hopefully we elaborate a more refined notation later):

a[nr]0=right neutral elment of [n]

a[nR]0=a

a[nl]0=left neutral element of [n]

a[nL]0=a

If we now start with the initial operation a[0]x=x+1, we have the following hierarchy:

a[1]x=a[0R]x=a+x

a[2]x=a[1r]x=ax

a[3]x=a[2r]x=a^x

a[3L]x=a[3L]x=a^a^x

and the following deviant operations:

a[1R]0=a, a[1R]1=a[1]a=2a, ...

a[1R]n=a(n+1)

There is no left neutral element e, e[1R]x=x => e=x/(x+1).

0 is the right neutral element

a[1Rr]0=0, a[1Rr]1=a[1R]0=a, a[1Rr]2=a[1R]a=a(a+1)=a^2+a, a[1Rr]3=a[1R](a^2+a)=a^3+a^2+a, ...

a[1Rr]n=

There is no left neutral element e, e[1Rr]x=x.

1 is the right neutral element

How can this be extended to the real n?

Deviant tetration

a[1Rrr]0=1, a[1Rrr]1=a[1Rr]1=a, a[1Rrr]2=a[1Rr]a=?

a[2R]0=a

a[2R]1=a a=a^2

a[2R]n=a^(n+1)

There is no left neutral element

0 is the right neutral element

Another deviant tetration:

a[2Rr]0=0, a[2Rr]1=a[2R]0=a, a[2Rr]2=a[2R]a=a^(a+1), a[2Rr]3=a^(1+a^(a+1)), ...

We saw already that the sub operations below [0] are all equal to [0], i.e. increments. And those are unfortunately the operation(s) that dont fit into this scheme. Because the initial value to go from a[-1]x=x+1 to a[0]x=x+1, is a[0]0=1, though neither 1 is a right neutral element of [-1] nor is 1=a.

But this can be reformulated that there are no operations below [0] that obey our general rule (*), i.e. the hierarchy starts at 0.