Few years back I found out that Burnout Paradise ultimate box demo has full game in it and to access it all you need was a product key (just like old big fish games or pop cap games). I wonder if there are more game demos which comes with full game with limitations.

Every Spiderweb Software game.

Basilisk Games' Eschalon series and, I think, Spiderweb Software's games are full version demos that are partially locked until you get the key. Not sure if there are retail versions (although I believe you can order discs from the devs)

Demigod also had something like that.

Master of Orion 2, if you patch the demo with the 1.31 patch, for some reason it turns into the full game. No idea why they did that.

DEFCON.

Jetboat Superchamps
<i>screenshots</i>

All the old Infinite Monkey Baseball/Football Mogul games definitely were full games activated via product keys.

Torchlight II (I think) - Runic's DRM, if you have a key, activate and you have the full game. Demo is limited to Act I.

Castlevania: Symphony of the Night (E3 Demo version) - According to TCRF, the save files are compatible with the Japanese version of the full game. You can bypass the demo limitations by playing the Japanese version and load the save files in the E3 version.

I think any game with time-limited trial, where you can try to get as far as possible before time runs out (I've played a time-limited Botanicula demo). If there is a way to crack the timer, then you'll get the full game (but that's not legal way).

If you patch the demo version of Tomb Raider Angel of Darkness then you get the whole game.

I believe Postal II demo was the full game but with a time limit.

You can download "Mount and Blade" and "Mount and Blade: Warband" from Taleworlds homepage.

The restrictions are described as follows on their download pages:

We are offering Mount&Blade through a try before you buy model. You can download the game and start playing right away. What you download is not a demo, it is the full game. There is, however, a level limit at 7. When your character reaches level 7, you can start a new game or, if you decide you like the game well enough, go on and buy a license online. As soon as you buy a license you will obtain a serial key which will remove the level limit. You will not need to download the game again.

A ton of 90s shareware used this model. I can't think of specific names ATM, though.

Few PS3 and PS4 game demo have same limitation as in Mount and blade.
Destiny trial and demo has same size as a full retail version with a limit of character growth to level 7. Although I don't know if it is a full game or not.

GOG Galaxy. You get to try the demo and eventually they'll patch it to full version.

What you're describing is called shareware. Just search for it, it was really popular for 'budget' games.