Clearly the winding number is $2$, because $2\pi$ worth of $t$ is one circuit around the origin, and there are two of them. This argument ought to be sufficient everywhere except in specialized classroom situations where you're required to use some particular method to determine the winding number. Since you're not specifying a method I assume this is not the case for you.
If you do want some kind of "more rigorous" argument, I recommend something like:
Let's construct a continuous argument function $f(t)$ for $\gamma$. Since $\gamma(0)=1$, let's set $f(0)=0$.
For $0\le t\le \pi$ the imaginary part of $\gamma(t)$ is nonnegative, so let's make $f(t)\in[0,\pi]$ in this interval. This leads, in some way we don't need to care about, to $f(\pi)=\pi$.
For $\pi\le t\le 2\pi$ the imaginary part of $\gamma(t)$ is nonpositive, so let's make $f(t)\in[\pi,2\pi]$ in this interval. This leads, in some way we don't need to care about, to $f(2\pi)=2\pi$.
For $2\pi\le t\le 3\pi$ the imaginary part of $\gamma(t)$ is nonnegative, so let's make $f(t)\in[2\pi,3\pi]$ in this interval. This leads, in some way we don't need to care about, to $f(3\pi)=3\pi$.
... and so forth ...