User blog comment:Primussupremus/A fast growing sequence of numbers./@comment-31663462-20170415152909/@comment-30754445-20170415171009

Change the first "<" to "≤" and that would be correct.

At any rate, it isn't too difficult to get much better bounds:

a1 = 2

a2 = 4

a3 = 256

a4 ~ 3.2317006071311007300714876689E616 ~ F2.44559482

a5 ~ E(1.9923739028520154087706422945E619) ~ F3.44589996

a6 ~ EE(1.9923739028520154087706422945E619) ~ F4.44589996

a7 ~ EEE(1.9923739028520154087706422945E619) ~ F5.44589996

a8 ~ EEEE(1.9923739028520154087706422945E619) ~ F6.44589996

and in general, for a≥5:

an ~ E(1.9923739028520154087706422945E619)#(n-4) ~ F(n-1.55410004)

(that's a power tower of n-4 tens, topped with a 1.9923739028520154087706422945E619)

The actual number on top would be exactly 256257 × log 256 for n=4. For n>4 this approximation would be correct to 616 significant digits.

Fun fact:

a259 is precisely equal to Steinhaus' "Mega" (2 in a pentagon in Steinhaus-Moser's polygon notation)