First off, betta are tropical
fish and need heaters unless you live in a very hot and humid place as they need temperatures above 76F.
The only way you can have a fish tank without some form of climate control (heater or chiller) is if you have a certain area of you house that stays a pretty fixed temperature year-round. For example: my goldfish tank is fine in the summer, but my house gets so cold in the winter that I need to have a heater to keep it above 70F. Fish need a constant temperature which is one of the reasons we use heaters. A tank without a heater will have more extreme temp fluxes which weakens the immune system of the inhabitants.
If you are looking at cooler-water species you could try White Cloud Minnows as Gwen suggested, but if your house gets too cold for them in the winter, you will need a heater. Another interesting species is the axolotl
, a species of neotonic salamander, that requires temps between 57F and 68F. Again, if it gets colder than that, you will need a heater, and they require a rather large and specialized setup.