There are two types: hyper^{4} and hyper_{4}.
The former, hyper^{4}, definition is used more often as an independent operator because only the other hyper4 operator can be reduced to double exponentiation in the form: a_{(4)}b = a^(a^(b1)), and thus is an extension of exponentiation. The hyper4 has not been extended to real numbers as addition (1st level dyadic operation), multiplication (2nd level) and exponentiation (3rd level) have.
Alternate names for hyper4 can include: tetration, superpower, superdegree, and powerlog.
This concept can be extended to the fifth degree and beyond as well, however each definition becomes recursive upon the last one... E.g.
Hyper4 is essentially equivalent to a↑↑b in Knuth's uparrow notation.
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