61.113 - is paying your pro rata share limited to the time you are in the plane?
The situation you're describing is a commercial operation. Without a commercial certificate (and a part 135 operation) you may not receive any compensation at all.

The FAA has consistently held that the 61.113's non-compensation exemption in 61.113(c) requires common purpose for the pilot and passengers (Bobertz - 2009, p.2):


  Absent a bona fide common purpose for their travel, reimbursement for the pro rata share of operating expenses constitutes compensation and the flights would be considered a commercial operation for which a part 119 certificate is required.


There is unfortunately not a strict definition of common purpose. However, the basic ground rule is that to fall under 61.113(c), the pilot must have their own reason for making the flight, independent of the passenger's.

The Bobertz interpretation presents an illustrative example: if a pilot wanted to go to a race, they could bring contestants to the race as passengers, and receive compensation. Because the pilot was flying there for the purpose of attending the race, common purpose is established, and pro-rata cost sharing is permitted.

However, if not all of the members of the race team could fit in the plane, the pilot could not share the cost to go back to the origin to make a second trip: there's no independent reason to do so. This crosses the line into a commercial operation, so no compensation is allowed.

Applying these guidelines to your question, the key phrase here is:


  I made all legs of the flight for the sole purpose of entertaining his desire.


There is clearly no common purpose. Therefore, accepting any compensation for the flight, even a pro-rata cost, is compensation in violation of 61.113.