How can I power a 3.3v circuit with a 9V power supply and two AA batteries?
The TPS63070 buck-boost converter accepts 2-16V, however in the comments you say it is 'cost prohibitive', so a cheaper solution is required.

Here is a module which sells on eBay for US$2.75, with free shipping!



Your power jack can switch between the battery and power adapter with a circuit like this:-







simulate this circuit &ndash; Schematic created using CircuitLab

The diode is not strictly necessary, but is cheap protection against reversed input voltage (which might happen if someone plugs in the wrong power supply).

Some other modules are even cheaper, but have a lower maximum input voltage (eg. 5V). For these you would need another regulator to reduce the 9V. As you only need 100mA a Zener diode shunt regulator might work (the Zener diode would be wired between the buck-boost regulator's input and ground, and D1 replaced with a resistor to limit shunt current). Or you could use a negative linear regulator such as the LM7905. However these solutions would increase the parts count, and might not be any cheaper overall.