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The standard trick is to use the logaritm in base 10.

Let x = lg(8600!). Since lg(xy)=lg(x)+lg(y) you have

x = sum(lg(k),k=1..86000)

Define floor(x) as the integer part of x. Then

86000! = 10^(x-floor(x))*10^floor(x).

You can evaluate x to the degree of accuracy you want, for example,
using Maple I got:

86000! = 7.91222558*10^372239

The compuation time was close to 0.