n! = n*(n-1)*(n-2)*...*3*2*1
n!! = (((...(((n!)!)!)...)!)!)!)! with n levels
n!!! = (((...(((n!!)!!)!!)...)!!)!!)!!)!! with n levels
n!!!...!!! = (((...(((n!!!..!!)!!!...!!)!!!...!!)...)!!!...!!)!!!...!!)!!!...!!)!!!...!! with n levels
n!2 = n!!!...!!! with n !'s
n!2! = (((...(((n!2)!2)!2)...)!2)!2)!2 with n levels
n!2!2 = n!2!!!...!!! with n !'s
n!3 = n!2!2!2 ...!2!2!2 with n !2's
n!! = n!n
n!!+1 = n!!!!!! ...!!!!!! with n !!'s
n!(!)! = n!!*(!-1)*...*3*2*1
n!!! = n!(((...((!)!)!...)!)!)! with n !'s
n!!!! = n!(((...((!!)!!)!!...)!!)!!)!! with n !!'s
n!!2 = n!!!!...!!! with n !'s
n!!! = n!!n with n !'s
Replace n!_! into n!^!, and n!_!_! into n!^(!^!)
Then:
n!^1^2 = n!^(!^(!^(...(^(!^(!^!)))...))) exponentiated n times
n!^1^3 = n!^1^(!^(!^(...(!^(!^(!^()^2)^2)^2)...)^2)^2) with n levels
n!^1^1^2 = n!^1^(!^1^(!^1^(...(!^1^(!^1^(!^1^())))...))) with n levels
n!^^2 = n!^1^1^1^...^1^1^2 with n ^1^'s
etc.
(sorry, too lazy)
-- UNDER CONSTRUCTION --