Finally someone who isn't caught by the dragnet of complicated BS surrounding this. The confusion by this is entirely created by DPEs trying to screw with people. There is nothing wrong with flying around in your friend's plane with or without your friend. The FAA doesn't want the flying public to have a misguided belief that that a PPL holds the same standards as a commercial flight...that's it.

There are two clear potential problems that everyone conflates, holding out and pro rata share. If there is no holding out then incidental compensation from getting to fly his plane for free doesn't matter. If there is no exchange of money on the trip then the tests for pro rata share are also irrelevant.

AC 61-142 has examples clear as day where it acknowledges that a flight can be done even without a common purpose, just that no expense sharing may take place. People seem to think if it fails the common purpose test then you can't do the flight, and that's just incorrect:

9.2...The common purpose test can be stated as “but for the receipt of compensation, the pilot would not have taken that flight.”

OP is taking the flight to help a friend and would do it even if the hours didn't count.

Example 1: A friend asks a pilot to fly him to another city to pick up a new car he ordered and offers to share the expenses of the flight. The private pilot agrees, as he is not doing anything else and would enjoy the flight. As the passenger chose the destination and the private pilot does not have a purpose of his own to be in the other city at that time, this is an example of a situation where no common purpose exists. Therefore, expense sharing would not be allowed under § 61.113(c).

It doesn't say you cannot do the flight, you just can't share expenses. OP should probably technically fill the tanks if someone is watching over their shoulder 🙄.

9.3.2 In a multiple-flight situation, the pilot may meet the common purpose criteria for expense sharing with some, but not all, of the passengers.

Example: Seven members of a running club, one of whom is a pilot, are participating in a race and they rent a four-seat aircraft to travel to the event. The pilot has to make two flights each way to transport all seven members. The pilot shares a common purpose with the passengers only in the first flight. This is because following the first flight the pilot would be at his destination and able to participate in the race. The second trip fails the common purpose test because the pilot has already achieved his purpose for the flight, i.e., getting to the race location to participate. Any additional trips would be solely for the reason of transportation of the additional members of the running club, and no common purpose would exist. Therefore, the pilot would only be able to share expenses with the first group of passengers.

This very clearly does not say the flight cannot be legally conducted just that expense sharing cannot take place. This also clearly lays out a path with the pilot getting incidental flight hours flying all the way back and forth on a second trip where they may not receive compensation.

The bit about flight time as compensation is really only relevant specifically in reference to holding out. They are trying to say advertising to people that you will take them anywhere they want to go, even if you don't collect money, doesn't mean you weren't compensated for the flight that you offered to the general public.

10.2 What Constitutes Holding Out? “Holding out” is accomplished by any means that communicates to the public that a transportation service is indiscriminately available to the members of that segment of the public that it is designed to attract. There is no specific rule or criteria as to how holding out is achieved. Instead, holding out is determined by assessing the available facts of a specific situation. Advertising in any form raises the question of holding out. Historically, pilots have been found to be holding out when advertising services via rolodex, brochures, newspapers, magazines, telephone directories, posters, and website/internet postings.

10.2.1 When money or anything of value is exchanged for transportation, the public expects, and the FAA demands, a higher level of safety for the flying public. A pilot may invite passengers for expense-sharing flights; however, a pilot should be guided by whether he or she is reaching out to a defined and limited group comprised of people with whom he or she has an ongoing, pre-existing relationship (e.g., family, friends, or close acquaintances). Generally, the FAA would not consider a mere loose acquaintance to be part of a defined and limited group, which is a principle that may have added relevance in the age of social media (see the discussion in paragraph 10.2.3.3.2).

10.2.2 The FAA distinguishes between offering expense-sharing services to a wide audience and to a limited group because holding out to the public may suggest to unsuspecting passengers that the pilot has met the higher regulatory requirements to carry passengers.Absent this limitation on holding out, an unsuspecting passenger may unknowingly assume the safety risks of flying in aircraft flown by pilots who lack the training, experience, and operational oversight that the FAA requires of operators that conduct common carriage.

The pilot's friend clearly does not assume OP has met higher regulatory requirements to carry passengers.