First, we must define what we mean by "a number we could count to". If we mean the largest number that we could count to in practice, in one lifetime, then yes, there would be a maximum such number. Even if we skip numbers while counting, there exist numbers large enough that merely saying their name would take longer than a human lifespan. For instance, even if we could pronounce (or think) one digit per millisecond, saying the first 10^15 digits of pi would take 30,000 years.
But that's just due to physical limitations of humans; it's not interesting mathematically. So let's define counting a bit more rigorously. We define the successor function S(n) = n + 1. We then start counting at 0, and the successor function gives us the next number we should count. For any natural number (non-negative integer) i, if we count like this long enough, we will eventually reach i, no matter how large.
But we will never reach a point where we've counted all the natural numbers. If we've been counting one number per second, then after i seconds, we'll be up to the number (i+1). But we can't stop yet; we haven't even covered (i+2), much less (i+42). We can count to any specific natural number in a finite amount of time. But we'd need an infinite amount of time to count all of them.