Some of the most frequent examples posted here:
1. Traffic light - Even though PLCs are not widely used for traffic light controllers. Just about everybody understands how a traffic light is supposed to work, so you don't have to spend a lot of time definining the task. It seems simple at first, until you get involved in the details. There are plenty of options you can add to make it more complex once you've mastered the basics.
2. Alternating between two devices, usually compressors. In this one, you are asked to come up with a program to control the operation of two compressors so that the usage is equally divided between them. Each time one of them needs to be started, the control alternates between the two. This example has many potential variations, so you have to be careful to define your objectives clearly.
3. Flip-flop logic. Sometimes defined as a one-button start/stop. Here you are asked to come up with logic such that actuation of an operator input causes an output to chnage state; if the output is off, actuation of the input turns it on; if the output is on, actuation of the input turns it off. Note that this one can be applied to #2 above.
4. Elevator control logic. Like the traffic light, everyone knows how an elevator is supposed to work. You can start with an elevator operating between two floors and expand your logic to cover additional floors.