You do NOT want to skimp out on a filter. Especially on a 10G tank. I understand no filters on 1G tanks, because daily water changes of 100% are no problem. If you don't have a filter on a 10G tank, you are going to have a ton of algae and waste buildup unless you change 50% of the water every day.
Stagnant water = bacterial breeding ground.
Filters are not expensive, especially for a 10G tank. I got my filter, new, for $10. Whisper 10i, and I have a betta in there ATM. Its great, and silent.