According to the following article,
http://www.aquariumcarebasics.com/freshwater-shrimp/amano-shrimp/ the amano shrimp grow to two inches. It also mentions they are likely to be eaten by aggressive tank mates, so again, it would largely depend on the individual betta. I would think you'd have a better chance to successfully keep these with female bettas since they're not usually the fighting gender, but I could be wrong on that.
On the same site I found info about bamboo shrimp (
http://www.aquariumcarebasics.com/freshwater-shrimp/bamboo-shrimp/), which mentions it eats micro algae (you can probably crush up an algae water for them), micro organisms, and other fine particulate matter. These do grow up to three inches, so just keep that in mind.
Again, same site mentions ghost shrimp (
http://www.aquariumcarebasics.com/freshwater-shrimp/ghost-shrimp/) which can grow to two inches, though 1.5 inches is more common.
You can click the above links for full info on the shrimps.
Now, my recommendation is to start with a couple ghost shrimp due to being fairly cheap. They're commonly used as feeder shrimp, so you can usually purchase them for a few cents. This will allow you to gauge the temperament of your betta, so you'll know if it's aggressive or not.
If all goes well, I may consider adding an amano shrimp or two, just to better gauge the betta's temperament with the new shrimp (remember, aggressive betta and the shrimp will end up like the rcs). However, PLEASE NOTE that shrimp do best in groups of at least six (the smaller "schooling" species, e.g. amano, rcs, ghost, etc. However, larger more solitary species are fine with three or four to an aquarium, e.g. bamboo shrimp), but you really only want to start with one or two so you don't lose too much money in case your betta eats them.
Alternatively, you could purchase a couple bamboo shrimp instead of amano shrimp. The most important thing is to start with the cheapest shrimp (ghost shrimp) and head to the other one, potentially the one you actually want, after you observe the shrimp happy and healthy.
Lastly, if you don't see any ghost shrimp after you put them in the aquarium, don't automatically assume they've been eaten. They're called ghost shrimp for a reason. I've owned some that I didn't see for days and one day I saw them again. However, they should be a good gauge of your betta's temperament to help determine which tank mates would be best for future ventures
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