What were the tallest mountain ranges in Earth's geological past?
Factors determining the maximum possible height of mountains include the rate of uplift versus the rate of erosion[a] and rock strength.

Rock strength is controlled by the type and internal structure of the rock in question. There is some evidence that once mountains extend above the snow line, glacial and periglacial erosion have a stronger control than exhumation/uplift rate (Brozovic et al, 1997; Egholm et al, 2009).

Everest and the Himalaya have reached their maximum possible elevation: the formation of the Tibetan plateau is due to the failure of rocks preventing the maintenance of discrete mountain peaks. The principle of uniformitarianism suggests that - subject to differences in variables discussed by Egholm et al, including crustal composition - the Himalaya and Tibetan plateau are an excellent approximation to the maximum achievable height of mountain ranges. However, identifying which specific palaeoranges were tallest (as opposed to calculating a plausible upper limit on height) is a significantly harder problem to solve.

[a] Though note that the rate of erosion increases as the rate of uplift increases - for more on erosional equilibrium, see e.g. Riebe et al (2000).