Banking.
Let me store multiple currencies. dont convert currencies but actually let me hold whatever currencies I regularly deal with ie Euros, US dollars, Australian dollars and Thai baht but it would vary customer to customer.
Let me make and receive payments over the net similar to PayPal (act as a payment processor).
Also give me a bank account number for people who want to pay via what they think is an old fashioned bank transfer. If you could provide me with whatever numbers I need so I can receive bank transfers in every country I do business that would be ideal.
I dont need any physical shop fronts. less real estate and staff should keep costs down.
Provide a net based interface and a card that works over the visa network and Im happy.
Do NOT send me paper statements and ads for your latest attempt to take my money under the guise of being my best friend.
The world could use a net-centric bank that assumes you are not limited to whatever country you happen to be physically located in right now. With its lack of physical storefronts this bank wouldnt be for everyone but thats fine. It would be a a bank for tech savy, international business people.