Which printer has the cheapest ink

First - don't get a printer with "combo" cartridges (that is, HP, Lexmark).
Second - get more expensive printer but with separate ink cartridges (Canon, Epson).
Third - check whether the cartridges the printer uses have chip on them, if you can - avoid this cartridge.

There are many places where you can get "remanufactired" or "compatible" cartrodges. YMMV (Your milleage may vary).