In the defining factors that make up (10^666)!, i.e., 1, 2, 3, ... , 10^666-2, 10^666-1, 10^666: 

Each multiple of 5 contributes a zero

Each multiple of 25 contributes a second zero

Each multiple of 5^3 contributes a third zero

...

Each multiple of 5^n contributes an nth zero

The number (10^666)! has as its defining factors 10^666 / 5 multiples of 5. 10^666 / 25 are multiples of 25, etc.

The last contribution is made by 5^952; that is 5^952 itself as 666/log(5)=952.83058767688. 10^666/5^952 is approximately 3.807, so there are only three multiples of this power present as defining factors in the factorial. The division by the next power of 5 is less than 1, so its floor function is zero.

So the number of trailing zeros would seem to be 10^666*(1/5 + 1/25 + ... )

However, not only is 10^666 not divisible by 5^952, as there's a remainder on that last division, but somewhere along the line (starting at 5^667, actually) there are previous powers of 5 that leave remainders in the division. So a simple geometric series formula won't work.

Use of UBASIC leaves errors during the divisions, where we need to take the floor of the division of 10^666 / 5^n. The INT function (floor function) is sensitive to the accuracy.

Mintoris basic on the Android cell phone allows as much precision as needed and indeed is faster than UBASIC interpreted under DOSBOX.

The Mintoris basic program

Open 2,"leviathan.txt","w"

nmrtr$=bigpow$("10",666)

tot$="0"

Print nmrtr$

for i=1 to 952

  tot$=BigFloor$(BigAdd$(tot$,BigDiv$(nmrtr$,BigPow$("5",i),1000)))

next

print tot$

Writeln 2,tot$

Close 2

performs the sum, finding

24999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999857

I've split the number into groups of 80 digits so this number is 8*80+26=666 digits long. It's 2.5 * 10^665 - 143.

Wolfram MathWorld corroborates this(http://mathworld.wolfram.com/LeviathanNumber.html) as

25 * 10^664 - 143 trailing zeros.