A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings. In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. This answers a question of Ying-Qing Wu.
"Totally geodesic surfaces with arbitrarily many compressions." Algebr. Geom. Topol. 11 (2) 643 - 654, 2011. https://doi.org/10.2140/agt.2011.11.643