There can't be a number
It doesn't make sense. Think about it: you're saying there are infinitely many zeroes, and then a one. So the one is after the last of all those infinitely many zeroes. But guess what? The whole point of infinity is that there is no "last" zero!! It's sort of like the line from The Phantom Tollbooth: "Just follow that line forever, and when you get to the end, turn left." You don't get to the end.
Does "the smallest number greater than zero" exist? Depends on your context. Obviously, in some contexts, like number theory, where "number" generally refers to natural number, there certainly is a smallest number greater than zero: 1. But that's obviously not under discussion here.
There is no real number which is the smallest number bigger than zero. CentrX asks how come it doesn't exist when infinity does. Well, there is no real number which is infinity either, so in a sense they're in the same boat. Infinity is not a real number, though it can be represented in other systems. And of course, there are uncountably many infinities of different sizes, which only confuses matters. Take the reciprocal of one of those infinities, say aleph null? Fair enough. You'll have to give that a meaning somehow, and whatever it is won't be a real number (though it may be meaningful in some other system). Is 1/aleph_0 bigger than 1/aleph_1? Um. Probably, if your system is coherent. In which case, which is really the smallest number? It would be the same as asking what's really the biggest infinity,. and just as unanswerable (though I've heard there are concepts of the "greatest" infinity, sometimes called "Omega" (with a capital Omega).
John Conway's "Surreal Numbers," as described in a book by Donald Knuth of that name, admit transfinite numbers, and even describe sensible ways they behave in arithmetic. So you can actually take the reciprocal or even the square root of infinity in his system, and it makes a certain kind of sense.
But in simplest terms, just as there is no largest number (infinity is not a number in the usual sense), there is also no smallest positive one.