Cheap windup alarm clocks or kitchen timers are good trigger devices and can do work, too. Obviously, they can be timing devices, too.
Candles are popular (but you can’t cheat and light the thing yourself; you need a trigger to strike a match or other flame to start the fire). The candle can be used for timing (a candle in a horizontal alignment will have its flame move along the length of the wick, and can burn through a string to trigger something happening, and (see below)
With the candle, you can do things with hoods to capture the hot gases and then rise as another trigger mechanism; depending on the sizes of the candle and hood (and its mass), this is another timing mechanism if you can judge how long it takes the hood to rise, for example.
All kinds of gravity devices are available, from falling dominoes (very popular) to balls rolling along inclined planes (a trough, so you can control its direction) to water devices, as others have mentioned.
The point of the exercise is two-fold: for you to recognize various types of mechanisms and triggers yourself, and then to be somewhat ingenious in making a series of them cheaply… and to be aware of the physics that make them work.
Don’t forget friction!