Vote Winner: n<m>3 = n^^^m
Pentation Numbers are Hyper-Operators Number part.
Multiplication Numbers: n(m)
Exponentation Numbers: n<m>
Tetration Numbers: n<<m>> = n<m>2
Pentation Numbers: n<<<m>>> = n<m>3
Hexation Numbers: n<<<<m>>>> = n<m>4
Heptation Numbers: n<<<<<m>>>>> = n<m>5
etc.
Pentation Numbers start with 10^10^10^10^10^10^10^10^10^10 and end with 10^^10^^10^^10^^10^^10^^10^^10^^10^^10.
Hexation Numbers start with 10^^10^^10^^10^^10^^10^^10^^10^^10^^10 and end with 10^^^^10.
Heptation Numbers start with 10^^^^10 and end with 10^^^^^10.
etc.
X<<X>> = 10^^10
X<<X<X>>> = 10^^10^10
X<<X<<X>>>> = 10^^10^^10
X<<X<<X<<X>>>>>> = 10^^10^^10^^10
n<<n<<n...n<<n>>...>>>> (m n's) = n<<<m>>>
X<<<n>>> = 10^^^n
X<<<X>>> = 10^^^10
X<<<<n>>>> = 10^^^^n
X<<<<X>>>> = 10^^^^10
X<<<<<n>>>>> = 10^^^^^n
X<<<<<X>>>>> = 10^^^^^10
etc.
n<<...<<m>>...>> (l <> brackets) = n<m>l
n<m>l = {n,m,l}
X<X>X = {10,10,10} = tridecal
Next is Expansion, Multiexpansion, Powerexpansion, etc. Numbers.