Convert numbers between 0 and infinity to numbers between 0.0 and 1.0

$\exp$ is a great tool, but there's also $$ x \mapsto \frac{x^2}{1+x^2} $$ which may be slightly easier to work with in some situations.

As @Servaes points out, you can also use $$ x \mapsto \frac{x}{1+x} $$ because you're working on the nonnegative reals rather than all reals.

And a personal favorite of mine is $$ x \mapsto \frac{2}{\pi} \arctan(x). $$

There are many options, one example is the function $f(x)=e^{-x}$. It maps the domain $(0,\infty)$ onto the range $(0,1)$, though it reverses the ordering. That is, if $x<y$ then $f(x)>f(y)$. This is of course easily fixed by taking $$g(x)=1-f(x)=1-e^{-x},$$ instead.