Do the auto-super-logarithm
Haskell, 55 54 52 bytes

s n=[x|x&lt;-[2,1.9999..],n&gt;iterate(x**)1!!floor n]!!0


Usage:

&gt; s 100
1.445600000000061


Thanks to @nimi for 1 byte!

Thanks to @xnor for 2!