What math operation comes after exponentiation (^)?

Question

Let's give expressions some names:

x+3 = x+1+1+1 // lvl 1
x*3 = x+x+x   // lvl 2
x^3 = x*x*x   // lvl 3

What would be the mathematical term / real name / topic for lvls after 3?

Like:

x (lvl 4 operation) 3 = x^x^x
x (lvl 5 operation) 3 = x(lvl 4 operation)x(lvl 4 operation)x
..

Thanks in advance.

P.S. Above are not programming codes/languages

Answer 1

Tetration is the next hyperoperator after exponentiation. The operator is noted as ↑↑ — or ^^ in ASCII — following Knuth's notation.

The next operation in the sequence is pentation, then hexation, heptation, octation, etc. Ackermann's three-argument function calculates these operations recursively.

╭────────┬────────────────┬─────────────┬───────────────────╮
│  Level │     Name       │  Notation   │ Ackermann (3-arg) │
╞════════╪════════════════╪═════════════╪═══════════════════╡   
│    1   │   Successor    │ a++ (unary) │   φ(a, 1, 0)      │
│    2   │   Addition     │ a+b         │   φ(a, b, 0)      │
│    3   │ Multiplication │ a×b         │   φ(a, b, 1)      │ 
│    4   │ Exponentiation │ a↑b         │   φ(a, b, 2)      │
│    5   │   Pentation    │ a↑↑b        │   φ(a, b, 3)      │
│    6   │   Hexation     │ a↑↑↑b       │   φ(a, b, 4)      │
│    7   │   Heptation    │ a↑↑↑↑b      │   φ(a, b, 5)      │
│    8   │   Octation     │ a↑↑↑↑↑b     │   φ(a, b, 6)      │
╰────────┴────────────────┴─────────────┴───────────────────╯