There are frequent arguments over exactly what constitutes an RPG. Rather than ignore or reject these, I will list out a number of cases for consideration:
1) A solo "Tunnels & Trolls" adventure: This is a solo gamebook consisting of a number of numbered paragraphs of text. Each will have instructions telling you which number to go to next based on your choices and possibly the result of die rolls. Unlike the choose-your-own-adventure books, the player has a full character sheet and uses the full RPG rules from the tabletop Tunnels & Trolls game. As a related case, you could consider an RPG combat system used for a wargame scenario.
For more examples of the variety in solo adventures, I would suggest Demian Katz's Gamebook Web Page.
2) The Extraordinary Adventures of Baron Munchausen: This is a storytelling game where each player makes up an outrageous 18th century noble and then they play a storytelling game in-character. Actions are limited to storytelling and a special form of duelling.
3) Free Kriegspiel: A old German term for player-moderated wargames, which are described in detail at Henri De Catt's Free Kriegspiel Pages. This is a wargame with a judge or judges who decide the result of moves and combats systemlessly, based on their knowledge of real-world military. Players may thus attempt anything they can think of, including unorthodox tactics, special actions, and so forth.
4) A freeform mystery game: like my own game The Business of Murder. Here the players all have full character information and are allowed to say whatever they like. The limit is that only dialogue is allowed (i.e. no touching, no actions beyond ones you would do in person).
5) A structured mystery game: like Decipher's "How to Host a Murder" series. Here play is broken up into rounds where information is handed out. The key difference from the former is that the players don't actually know what their characters do. The person who turns out to be the killer may not know that until a later round. Really it is more of just a mystery-solving game with some added color. Acting like your character really isn't an intended part of the game.
6) Once Upon A Time: This is a storytelling card game, where players take turns telling a story, and they can lay down cards if they convincingly work the named element into the plot. There is no association of player with character, but it is a game which forms a story.
My own personal definition is whether a player might comment "No, my character wouldn't do that." -- i.e. considering imagined fictional personality as a real limit. Based on this, I suspect that a Tunnels & Trolls solo adventure is not generally an RPG since I don't think that people consider character as a limit on actions. Baron Munchausen is borderline, but seems like an RPG in that the generated character changes what stories are told. Free Kriegspiel is not generally an RPG because the commander is not considered as a character. The same procedure could be used for an RPG, however. From this logic, a freeform mystery is an RPG but a structured mystery is not. Once Upon A Time is not an RPG, then, by my definition.
However, that is fairly arbitrary. There is no definitive answer to this. There are many ways of breaking up these cases.