The voltage in a battery is determined by the chemical reaction that creates the potential difference between the cathode and the anode. The voltage formed by the reaction is called the terminal voltage of the cell/battery. In alkaline and heavy-duty batteries which are the two types of consumer electronic non-rechargeable batteries that you can find in stores in the usual sizes, the terminal voltage is 1.5V. A battery made from a zinc-galvanized nail and a copper penny stuck into a lemon has a terminal voltage of about 1V. A lead-acid battery - such as used in cars and UPS's - has a terminal voltage of 2.1V (6 are stacked in series in a "12V" car battery).
The two most common types of rechargeable batteries that come in common consumer electronic sizes are nickel-metal-hydride and nickel-cadmium batteries. They have a terminal voltage of 1.2V. On the other hand, in applications where weight and size are important, lithium-based rechargeables are taking over - such as lithium-ion, lithium-polymer - and they have a terminal voltage of 3.6V. They aren't made in common consumer sizes because they would fry the electronics since the base voltage on them is more than twice that of a typical consumer cell.
Since the terminal voltage is a function of the materials and the chemistries involved, there's not much that can be done to change the base voltage without abandoning the battery technology completely. So, if you have a zinc-carbon battery, you are stuck with 1.5V. If you have a nickel-cadmium battery, you are stuck at 1.2V.