You could go with a leopard frog, or maybe firebelly toads.. but FBT can NOT be handled at all; not even while cage cleaning unless you have rubber gloves on. Their skin contains a toxin that is very irritating. My now-husband, then-fiancee picked one up once,, itched his eye afterward.. his eye turned beet red and burned.
Both of these will eat mostly crickets but need diversity in their diet; earthworms, etc.
That said, they are absolutely fascinating animals to watch. They are not at all handling pets; the oils in our skin aggrevate them and mess up the balance of their skin; see, they absorb water through their skin rather than lapping it up like other animals. I would recommend wearing rubber gloves anytime a frog needs to be transported (from tank to holding container, for tank cleaning etc..) to protect the frog from the oils on your skin.
You could go with 100% aquatic frogs like dwarf or African clawed frogs; they are exactly like having fish in that you do not handle them; they are fully aquatic; you could have one in a 10g tank for its life but it needs a secure lid; they can get out! I have 5 of these in a 125g tank amongst other things and they are so entertianing.