If you model a battery as an ideal voltage source in series with a resistance, then putting batteries in series will increase the open-circuit voltage by n times the number of batteries in series, but the short-circuit current will not change because the internal resistance also increases by n times.
For more moderate loads than a short circuit the current will increase with the number of batteries. For example, if your battery has a 1.5V voltage and a 1 ohm source resistance and you connect 100 in series you will have 150V and 100 ohms source resistance. Connecting them to a 1000 ohm resistor will give you 136mA (150V/1100 ohms). One battery across 1000 ohms will result in a current of a bit under 1.5mA (1.5V/1001 ohms).
Batteries don't quite behave like voltage sources with fixed resistors in series-- the internal resistance changes with battery condition, temperature and history of discharge, but it's a reasonable first approximation.