The answer came to me as I was writing the last sentence of the question: It's the Well World series by Jack L. Chalker (what I remembered, oddly, wasn't "Well World" but "Well of Souls," part of the title of the first novel, Midnight at the Well of Souls). The "old man" is Nathan Brazil.
From the Wikipedia link:
The Well World series is a series of science fiction novels by Jack L. Chalker. It involves a planet-sized supercomputer known as the Well of Souls that builds our reality on top of an underlying one of greater complexity but smaller size. The computer was built by a now extinct race, the Markovians, who developed the Well of Souls with the goal of creating a new species that would transcend their own.
The Well World is the planet that houses the Well of Souls, and it exists within the original Markovian reality. Its surface was used as an experimental site where the Markovians tested their species designs before sending the successful ones into the new universe to populate planets. Humans are one of many such designed species who now live in the "real" world. During the time period of novels, the Well World has been abandoned and left on its own for an unknown length of time.
The books mainly follow a mysterious character known as Nathan Brazil, who has an (initially) unknown connection to the Well World. The books are adventures that follow Brazil and a changing cast of secondary characters through a series of visits to the Well World over a period of hundreds (and millions) of years.