Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, and I am interested to know whether there also exists an exponential upper bound. If we restrict to alternating knots, then work of [Lackenby](http://arxiv.org/abs/math/0012185) is relevant.



How many knots are there with hyperbolic volume less than a given constant
Here is an expansion of Ian's answer. 


Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of [Thurston's book](http://library.msri.org/books/gt3m/PDF/). Furthermore, the new manifold has smaller hyperbolic volume than the original. See Theorem 6.5.6 in chapter six. Some further discussion and references can be found in Ian's answer [here](https://mathoverflow.net/a/133890/1650).


So: fix a hyperbolic link $L$ in the three-sphere with an unknotted component $C$. Surgering $L$ along $C$ gives an infinite family of hyperbolic links, all with volume less than that of $L$. For explicit pictures of this, see the last few pages of chapter five of Thurston's book. (Actually, the pictures go in the opposite direction -- he explains how the Whitehead link is the "geometric limit" of a sequence of twist knots.)


