A fourth level dyadic mathematic operation involving exponentiation of exponents.

There are two types; hyper4 and hyper4.

hyper4 is symbolized by either a^^b or a(4)b, and is defined as a(4)b = a^(a^(...^a)).

hyper4 is symbolized by a(4) and is defined as a(4)b = ((a^a)^...)^a.

The former, hyper4, definition is used more often as an independent operator because the other, hyper4 operator can be reduced to double exponentiation in the form: a(4)b = a^(a^(b-1)), and thus is not really any different from 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...
I.e. hyper5 = a^^^b = a(5)b = a(4) a(4)...a(4)
or hyper6 = a^^^^b = a(6)b = a(5)a(5)...a(5)
For more information on higher dyadic and triadic operators, see:

P.S. If someone more knowledgeable wishes to rewrite this, feel free to do so.
This writeup is under the GNU Free Documentation License