Can we make a rough estimate of the number of generations since the origin of life?
If you're willing to accept many orders of magnitude and define life as the Last Universal Common Ancestor.

For the rest of this answer, life begins 3.5 Gya with cyanobacterial mats and stromatolites and so on. Genetically the LUCA is dated to around this time, which matches the fossil record and everything's great. The LUCA can't have sprung from nothingness, so there may(will) have been (many)generations before the LUCA of things we would recognize as life, but I'm ignoring them. If they wanted to be considered they should have left descendants.

The fastest doubling times for modern bacteria are about 7-9 minutes but most are longer. E. coli takes about 20 minutes, and yeast takes hours. Given the 3.5 billion year clock, that gives an upper bound of about 1x10^14 generations. (Green sea turtles take 20-30 years to reach sexual maturity, or 1.1x10^9 generations). Something a bit representative is probably Pelagobacter ubique, which is vastly successful and takes about 29 hours to divide. Averaging out ice ages and so on and skipping the slowing of generations multicellularity probably implies(the Cambrian explosion is 82% of the way to the present day, being only 543 million years ago. We could account for it, but our guesses on generation time are way more inaccurate than any possible effect that could have) would estimate about 1x10^12 generations.