**The title and question have been edited in light of Ian Agol's comment. The previous question was stated in terms of the wrong notion of length to discuss deformations:** What is the longest slope $\gamma$ in the complement of Dehn surgery space of a cusped hyperbolic 3-manifold $M$? Here Dehn surgery space is the space of fillings such that the hyperbolic structure on the filling $M(\gamma)$ can be realized as a deformation of the original $M$. To keep things focused, consider only fillings of one cusped manifolds. This question is related to Ken Baker’s question: [Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds][1] However, Ken’s question is interested in the total number of slopes in this complement. This question is focusing on the longest such slope measured using normalized length, where the normalized length of $\gamma$, $\mathcal{L}(\gamma)$ is as defined in: <cite authors="Hodgson, Craig D.; Kerckhoff, Steven P.">_Hodgson, Craig D.; Kerckhoff, Steven P._, [**Universal bounds for hyperbolic Dehn surgery**](http://dx.doi.org/10.4007/annals.2005.162.367), Ann. Math. (2) 162, No. 1, 367-421 (2005). [ZBL1087.57011](https://zbmath.org/?q=an:1087.57011).</cite> Namely $\mathcal{L}(\gamma)=length(\gamma)/\sqrt{Area(\partial T)}$. Here, $length(\gamma)$ the translation length of $\gamma$ in a cusp neighborhood and $Area(\partial T)$ is the area of that torus in cusp neighborhood. Of course, it is possible that this question as stated does not have a realizable answer, because there is no longest slope. Here is a more carefully stated version: **What is the largest $\mathcal{L}_{max}$ such that there exist a family of slopes $\gamma_i$ in (1-cusped hyperbolic) manifolds $M_i$ such that $$\lim_{i \to \infty} \mathcal{L}(\gamma_i) = \mathcal{L}_{max}$$ and each $M_i(\gamma_i)$ is a hyperbolic manifold such that the hyperbolic structure cannot be realized as a deformation of the hyperbolic structure of $M_i$?** Hodgson and Kerckhoff give an upper bound of $\mathcal{L}_{max}\leq C\approx 7.515$. To give context how normalized length affects length, the (3,3,3) pretzel knot has slope of length 6 yielding a torus filling. However, the normalized length of this slope is $\mathcal{L}=\frac{6}{\sqrt{A}}\approx 1.91673$, $A=\frac{8\sqrt{3}}{(1+3\sqrt{57})^{1/3}}+\sqrt{3}(1+3\sqrt{57})^{1/3}$. This example appears in <cite authors="Adams, Colin; Bennett, Hanna; Davis, Christopher; Jennings, Michael; Kloke, Jennifer; Perry, Nicholas; Schoenfeld, Eric">_Adams, Colin; Bennett, Hanna; Davis, Christopher; Jennings, Michael; Kloke, Jennifer; Perry, Nicholas; Schoenfeld, Eric_, [**Totally geodesic Seifert surfaces in hyperbolic knot and link complements. II**](http://dx.doi.org/10.4310/jdg/1207834655), J. Differ. Geom. 79, No. 1, 1-23 (2008). [ZBL1158.57004](https://zbmath.org/?q=an:1158.57004).</cite> and as a filling of the manifold defined by Figure 6: <cite authors="Agol, Ian">_Agol, Ian_, [**Bounds on exceptional Dehn filling**](http://dx.doi.org/10.2140/gt.2000.4.431), Geom. Topol. 4, 431-449 (2000). [ZBL0959.57009](https://zbmath.org/?q=an:0959.57009).</cite> However, the cusp area appears to grow for (n,n,n)-pretzels, so I chose the (3,3,3)-pretzel to seemingly get the longest slope (in terms of normalized length) in this family. [1]: https://mathoverflow.net/questions/121774/hyperbolic-exceptional-fillings-of-cusped-hyperbolic-3-manifolds