It is possible to run a planted tank without a filter. However lets look at why using a filter is generally still recommended in a fully functional planted tank.
Many filters look great on the outside, and look really cool. But poor flow/out contributes to algae issues in a tank, and I would avoid these 2 types...
Choosing a tank for your initial planted aquarium setup is like picking a computer motherboard. It is a key "fixed parameter" that is generally difficult to change later. While folks often advocate that larger tanks are more stable - I hold a different opinion that focused effort in a smaller tank almost always creates better aquascapes.
How big a tank should I go for? If this is your first tank, smaller is not necessarily limiting. Here are some examples of great 'small' or nano planted tanks ranging from 7 gallons to 10 gallons.