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Rewriting the factorial as the Gamma function and Stirling's approximation we get what I think is the closest possible approximation that you could do by hand:
Where $e = 2.71828\dots$. Unfortunately, this might not be quicker than multiplying all the numbers together by hand, but it's certainly the only shortcut I can think of that could be done by hand.
I see that this is an old question and the conversation thread is most likely completely dead. However, there is a partial shortcut to calculating fairly length factorials. Being that multiplication is commutative, the order in which multiplication is done can be rearranged...
Take, for example, 10! which equals 1*2*3*4*5*6*7*8*9*10, this can be rearranged as (10*1)*(9*2)*(8*3)*(7*4)*(6*5) → 10*18*24*28*30
One thing that you will (or should) notice about the delta between the product of each pairing is that it always decreases by a value of 2, and always begins with a value of n-2. For n!, where n=10, the delta between the first two products will be n-2, or 8. The delta between the products of successive pairs will always decrease by 2. In other words, when the pairing is done as illustrated above (pairing the highest value number with the lowest, second highest with the second lowest, etc), then the product of each pairing will increase by a value that decreases by an amount of 2 for each successive pairing. This makes calculating the products of a long list of number pairs relatively easy.
For n!, where n=10, we know that 1*10=10, we also know that the next pairing will result in 10+8, or 18... the next paring will result in a product of 18+6=24, followed by 24+4=28, and finally by 28+2=30.
Without having to calculate each product, we can quickly predict what they will be.
This works for factorials of odd numbers as well, except that there will be one number at the end that will not have a pairing. 7!, for example, could be written as (1*7)*(2*6)*(3*5)*4 → 7*12*15*4. The initial product is 7, and the product of the next pairing is 7+(n-2) or 7+5, which of course equals 12. The next product is 12+3, or 15. So the solution to 7! would be 7*12*15*4.
As a generalized rule, it can be said that the initial delta between the products of the first and second pairings will be n-2, this delta will always decrease by a value of 2 for each successive pairing, and the final delta will always be either 2 or 3 (depending on whether we are dealing with the factorial of an even number or an odd number [2 or greater]).
Again, this is only a partial shortcut. It essentially reduces the number of calculations you need to perform by half... but if you absolutely need to do calculate 100! by hand, performing approximately half of the necessary steps would make a significant difference. :)