CJam, 2 × 10^268,435,457

A28{_*}*K*

This computes b, defined as follows:

• a₀ = 10

• a_n = a_{n - 1}²

• b = 20 × a₂₈

$ time cjam <(echo 'A28{_*}*K*') | wc -c
Real    2573.28
User    2638.07
Sys     9.46
268435458

Background

This follows the same idea as Claudiu's answer, but it isn't based on it. I had a similar idea which I posted just a few minutes after he posted his, but I discarded it since it didn't come anywhere near the time limit.

However, aditsu's suggestion to upgrade to Java 8 and my idea of using powers of 10 allowed CJam to calculate numbers beyond the reach of GolfScript, which seems to be due to some bugs/limitations of Ruby's Bignum.

How it works

A    " Push 10.                                                          ";
28{  " Do the following 28 times:                                        ";
  _* " Duplicate the integer on the stack and multiply it with its copy. ";
}*   "                                                                   ";
K*   " Multiply the result by 20.                                        ";

CJam, ≈ 8.1 × 10^1,826,751

KK,{)*_*}/

Takes less than five minutes on my machine, so there's still room for improvement.

This computes a₂₀, defined as follows:

• a₀ = 20

• a_n = (n × a_{n - 1})²

How it works

KK,   " Push 20 [ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ]. ";
{     " For each integer in the array:                                 ";
  )*  " Increment it and compute the its product with the accumulator. ";
  _*  " Multiply the result with itself.                               ";
}/