>>3147
What you're looking for is called tetration or power towers.
http://mathworld.wolfram.com/PowerTower.html
I tried using the Knuth notation a^^k, didn't work. I tried using PowerTower[a,k], didn't work. But apparently Nest[a^# &, 1, k] works. What Nest[f, expr, n] does is it repeats the function f n times on expr.
Take note though, 2^^1 = 2, first number in your sequence is already 2^^2.
You can also use Table[expr, n] to get out a table of values;
Table[Nest[2^# &, 1, n], {n, 1, 5}]