There are simple examples, and quite complex examples.
You could plot stars, with a vector in the direction you see the starlight from, and a length telling you the brightness. You are simply using a 3d vector to represent 3 bits of information (two for the direction, one for the brightness), and I don't know any way to add them sensibly.
Another example would be if you rescaled everything, maybe you had some very large vectors and some smaller vectors (but all bigger than unit length) and you want to see them all in one picture. Then you might erase all your original vectors and then draw new vectors that point in the same direction but that have a log of the actual length. You can see them all better now in the same picture, but don't use the parallelogram rule to add the rescaled ones, you use the parallelogram rule to add the originals, and draw the rescaled version of the original sum.
If that last example seems totally ad hoc, consider a related real life example. When you consider speed in relativity, they are all between $0$ and $c$, but sometimes we keep track of speed in a form called rapidity. If you wanted vectors whose directions are the direction of velocity and whose magnitude is the rapidity then you can compare the addition of those vectors to to the addition of velocities (either a Galilean addition or relatvistic addition). This is a bit of a hybrid of the two above approaches, you have a 3d vector with a direction in the direction (in space) of motion, and a magnitude that tells you something about the speed (either the raw speed $v\in[0,c]$, or the rapidity). In 1D, you can make it so that rapidity adds as a vector, where the sum of two rapidities (and each rapidity is the rapidity of one frame relative to the previous one). I'm not sure whether that addition is associative, however, it seems like I have to interpret each rapidity in regards to a pair of frames.
