from the two times i bought mattresses and one time I sold one:
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You at least used to be able to get a half-decent memory foam mattress for a few hundred bucks from online retailers. Since they can roll 'em up and vacuum seal them, it's easy for them to transport them. These are like 3-4 inches of memory foam, no box spring, but still decently comfortable especially if you're aiming to save up for a better one eventually.
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Look for factory-direct local mattress retailers. May or may not exist near where you are, but in a larger city you've got a shot. They're like half the cost vs mattress firm or similar stores and ime the quality is better too.
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Half a step above the shitstained road mattress: You could check to see if the people moving out of your new place are interested in selling theirs to you — since mattresses are hard to resell they might not plan on keeping it