Bettas can do fine in tanks as small as .5 gallons with daily (and i mean EVERY day water changes). In fact, they can even prosper in smaller quarters (thai breeders often raise their stock in 250 ml whiskey flasks), but it significantly increases the amount of work that you have to do to keep them healthy. If you are doing weekly water changes, you definitely need a filter. If you don't want a filter, start with the daily changes. Either way, your next betta will be much happier if you arm yourself with as much research as you can.