User blog comment:King2218/Infinite Extensibility/@comment-25371339-20140922184701

My Extension:

a↑ _1 b = a↑b

a↑ _2 b = a↑↑↑...↑↑↑a with b arrows

a↑_2 ↑_2 b = a↑2↑↑↑...↑↑↑a with b arrows

a↑_2 ↑b = a↑_2a↑_2a↑_2...a↑_2a↑_2a↑_2a with b a↑_2's

a↑_3 b = a↑_2 ↑_2 ↑_2 ↑_2... ↑_2 ↑_2 ↑_2 ↑_2  a with b ↑_2's

a ↑_4 b = a↑_3↑_3↑_3↑_3...↑_3↑_3↑_3↑_3 a with b ↑_3's

More coming soon!