One option that won't work for every situation is to take lots of short exposures and combine them in post-processing. You don't need a tripod for this because you can align the images in software before you combine them. Just shoot the images hand-held on burst mode, keeping the camera as steady as you can.
You can't get things like light trails from this method, but you can get a smooth "glassy" surface on water and turn crowds of people into diffuse shadows, similarly to the results you'd get from a long exposure.
If you have Photoshop, the procedure is to open your image files as layers, then select all the layers, and then select "auto-align layers" in the edit menu. This will line up all the layers perfectly, and you can then average them all together using one of the techniques described here.
If you can't afford Photoshop and don't mind hacking at the command line, you could use a utility called align_image_stack to align them, and then merge them using (for example) ImageMagick. (There may be more user friendly free software that can do it as well, but those are the ones I've used.)
If the camera moves around while you're shooting, you'll find that you can't use the edges of the final image, because they weren't covered by all the shots. So when shooting this way you should zoom out a bit more than you normally would, and be prepared to crop later.