User blog comment:Googleaarex/Extended Up Arrow Notation/@comment-10429372-20131128190550

This is nothing new.

$$a\uparrow_2b = \{a,a,b\}$$

$$a\uparrow\uparrow_2b = \{a,a,1,2\}$$

$$a\uparrow\uparrow\uparrow_2b = \{a,a,2,2\}$$

$$a\uparrow_2\uparrow_2b = \{a,a,b,2\}$$

$$a\uparrow_2\uparrow_2\uparrow_2b = \{a,a,b,3\}$$

$$a\uparrow_3b = \{a,a,a,b\}$$

$$a\uparrow_4b = \{a,a,a,a,b\}$$

$$a\uparrow_aa = \{a,a+1(1)2\}$$