# Julia, <s>34</s> 31 bytes

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    n->find(digits(prod(1:n)))[1]-1

This is an anonymous function that accepts any signed integer type and returns an integer. To call it, assign it to a variable. The larger test cases require passing `n` as a larger type, such as a `BigInt`.

We compute the factorial of `n` (manually using `prod` is shorter than the built-in `factorial`), get an array of its `digits` in reverse order, `find` the indices of the nonzero elements, get the first such index, and subtract 1.

[Try it online!](http://julia.tryitonline.net/#code=Zj1uLT5maW5kKGRpZ2l0cyhwcm9kKDE6bikpKVsxXS0xCgpmb3IgdGVzdCBpbiBbKDEsMCksICg1LDEpLCAoMTAwLDI0KSwgKDY2NiwxNjUpLCAoMjAxNiw1MDIpXQogICAgcHJpbnRsbihmKGJpZyh0ZXN0WzFdKSkgPT0gdGVzdFsyXSkKZW5k&input=) (includes all but the last test case because the last takes too long)