I have given a number of young people their first inspirational flight. I am a CFI, but I do not issue any endorsement for these flights, and do not charge for my time or the aircraft. Is there any FAR defining "introductory" or "familiarization" flight, or addressing how many times a non-flight-student (no student pilot medical certificate and no student pilot log book) can receive an introductory or familiarization flight?
There's no formal definition or regulation that I could find; introductory flights aren't defined in 14 CFR 1.1, for example. I'm not sure why the FAA would care how often you fly with someone, or what the difference is between an introductory flight and any other flight with a non-paying, 'non-logging' passenger.
The only other people who might care about it are the TSA but if you aren't providing instruction during the flight then that shouldn't matter either. Introductory flights are usually understood to be a "demonstration flight for marketing purposes" (49 CFR 1552.1) and therefore exempt from TSA requirements, but there doesn't seem to be any rule about how many demo flights you can make with an individual.
There is no FAR regarding intro flights for prospective student pilots. Different instructors handle things differently.
I use a log sheet which was created with a spreadsheet program, which I use for student flights. It has critique blocks for different parts of the lesson, which includes preflight, engine start, take-off, medium bank turns, etc. You get it.
The prospective student gets graded same as if they were a continuing student. I sign the sheet, and they do as well, and they get the original and I keep a copy. Their time is dual-received and mine is dual-given. They get the same post-flight briefing I give students, and it includes tips on handling parts of the flight, perhaps motion induced feelings, and what to expect in future lessons. I tell them to keep the sheet, and should they take up lessons in the future, that sheet can become an entry in their logbook, and put into an envelope in the rear of the logbook.
This way students have a better idea how lessons will work, and the same importance is given to the first lesson as will be the case in all subsequent lessons. It is situational, but usually my intro flights take the same 90 minute time that I plan for other training flights.
I always try to have the student practice a few things they do well at, such as power settings, trim or medium bank turns to a heading, so that there are parts of their critique which will show higher scores. The scores are genuine, and if the student does poorly, that is the critique they get, along with any coaching.
Through the years I have given initial rides to 75 yo and 9 yo. These rides are lessons, and not "Eagle Flights" or some other ride. Accordingly, I handle the TSA required documentation as if the flight initiated flight lessons.