Abelenky's answer is right, but the relevant regulation is FAR 91.409(b):
(b) Except as provided in paragraph (c) of this section, no person may operate an aircraft carrying any person (other than a crewmember) for hire, and no person may give flight instruction for hire in an aircraft which that person provides, unless within the preceding 100 hours of time in service the aircraft has received an annual or 100-hour inspection and been approved for return to service in accordance with part 43 of this chapter or has received an inspection for the issuance of an airworthiness certificate in accordance with part 21 of this chapter. The 100-hour limitation may be exceeded by not more than 10 hours while en route to reach a place where the inspection can be done. The excess time used to reach a place where the inspection can be done must be included in computing the next 100 hours of time in service.
So here are the rules:
- No person may operate an aircraft carrying any person for hire (for hire here refers to the pilot, not the passenger, and the person operating the aircraft is the pilot)
- No person may give flight instruction for hire (here person refers to the flight school or an instructor which the flight instructor or flight school provides) without a 100 hour inspection.
This means you can hire a flight instructor to instruct in your own aircraft without a 100-hour inspection. You can also fly a rented aircraft without a 100 hour so long as that flight is not under the guidance of a flight instructor.
Since you were not being paid, and you were not being instructed, the 100 hour inspection does not apply. However in practice I don't know of any aircraft rental service that allows the planes to fly beyond 100 hours regardless of it being an instructional flight or a private one.