I didn't see the post the other person wrote about cycling a tank
without a filter. Most of the bacteria in a cycled tank live in the
filter media so I'm not sure why they would advise you to do this. If
you're going to cycle your tank you need a filter. You'll be surprised
how fast two bettas will create ammonia in a 10 gal tank. Of course
each betta produces ammonia at a different rate and there are many
factors that lead to the build up of ammonia (filtration, type and
quantity of food, plants, etc)

Don't fall for the plant absorption method, it doesn't work. I can tell
you with 100% certainty that your plants are not going to be enough to
keep toxins out of the water. If you're going to try a tank cycling
experiment with any fish, (which I don't advise) bettas are not the
right choice. They are far to susceptible to the effects of toxins in
the water.