Ally Bank offers one checking account, one savings account, and one money market account, none of which charge a fee or require a minimum deposit amount.
"There are no hidden fees, and the savings account yields 1.05% no matter what your balance. The money market deposit account yields 0.85% on all balances and comes with a debit card and free checks. Even the checking account pays 0.1% on balances under $15,000 (and 0.6% on deposits above that)," according to Kiplinger.
Best internet bank runner-up: Bank of Internet
Best bank for millennials runner-up: Langley Federal Credit Union