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I would say invest in a ball as they are so versatile. A mat is always useful for doing exercises on (prevents rug burn or softens hard surfaces). Resistance bands are great if you don't have any dumbbells.
These would have been my suggestions as well.
Also, if you have a couple of 1/2 gallon plastic milk cartons, you can fill them with water or sand (1/2 full, all the way full, whatever) and they serve as pretty decent weights for things like curls and presses. It's nice to use the milk cartons because they have handles that make it easier to hold. Also if you don't fill them all the way, the weight shifts as you lift them, so you get kind of the effect of kettlebells. And a milk carton filled with sand is much cheaper than a kettlebell!
Also just basic body resistance stuff is good - pushups, tricep dips off a dining room chair, stepups onto a dining room chair, squats, lunges, all kinds of crunches and leg lifts, etc. You don't need any special equipment for any of that.
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