Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold.  Moreover, the core curve of the filling solid torus in these fillings is a geodesic in the resulting hyperbolic manifold.
Typically the study of exceptional Dehn surgery on hyperbolic 3-manifolds is concerned with the production of non-hyperbolic 3-manifolds.  However, it is possible for a hyperbolic manifold to result without being a deformation of the original.  For example, a "random" knot in a hyperbolic 3-manifold won't be isotopic to a geodesic though is likely to have hyperbolic complement.  The meridional filling of this knot exterior would then be an exceptional hyperbolic filling.
What's the maximum number of possible exceptional hyperbolic fillings? What manifolds realize this maximum?
 A: Sixty is an upper bound.
Hodgson and Kerckhoff's Universal Hyperbolic Dehn Filling theorem ("Universal bounds for hyperbolic Dehn surgery." Annals of Mathematics. 162(1), 367-421) says that, in a one-cusped manifold, you can push the cone angle up from zero all the way up to $2\pi$ as long as the normalized length of the filling slope is at least $7.515$.  This implies that the Dehn filling is a hyperbolic Dehn filling (meaning the core is geodesic) when the slope has normalized length that big.  This excludes at most 60 slopes.
A: I'll embellish on Autumn's answer. Experimentally, it seems that most one-cusped hyperbolic manifolds have at most 10 exceptional Dehn fillings in the sense you consider, i.e. points in the complement of Dehn surgery space. It is conjectured that Dehn surgery space is star-like, so that one can deform any hyperbolic metric with core geodesic by decreasing the cone angle monotonically to zero, and without affecting the singular structure. Lackenby and Meyerhoff have shown that there are at most 10 non-hyperbolic Dehn fillings, but they are not able to show that these are deformations of the complete metric in the sense of Thurston. However, their method of proof gives a bit more. The core of the Dehn filling is homotopically non-trivial, and  the kernel of the Dehn filling map on $\pi_1$ is a free group (this is true if the core is geodesic, since the fundamental group of the complement of a collection of geodesics in hyperbolic space is free). I'm not sure if they prove this in the paper, but it follows from the proof the 6-theorem. The number of negatively curved fillings is also much smaller than 60 from the $2\pi$-theorem. There's hope that the cross-curvature flow could flow these negatively curved metrics to the hyperbolic metric. If one could do this in the cone-manifold context, then this might enable one to obtain a sharpening of Hodgson-Kerckhoff. 
