Sure this can be done, to a degree. Lots of things like cell phone lighting do this (put the LEDs in series and drive from a lower voltage battery). This gets progressively harder to do as you start looking for higher and higher voltage outputs voltage wise.
"Back of the envelope" calculations are easy, get out your envelope.... Think in terms of power, from output to input. Let's say 20 mA per LED? At a typical 3.6 Volts this is (20 times 3.6), 72 mW mili Watts since we were using mili Amps (rather than whole Amps, where we get answers in Watts...). Times say 4 LEDs in series (or parallel for that matter power is power...) gives us 288 total, call it 300 mW. This power needs to come from the battery, through our converter.
If the converter needs to deliver this power to the load (the LEDs), it needs to draw it and more (efficiency) from the battery. We need to estimate efficiency, sure to raise a lot of strong opinions. For now, let's say 70%, near the top for what I consider realistic ranges (say 50 to 90%, depending). So we need to divide out 300 mW by 70% (.70), meaning we need 428 mW from the battery.
Let's take a first pass at the battery as a single
Energizer E91 AA cell. Which we can assume at 1.2 Volts to start.
If we divide our power need, 428 mW, by this voltage, we get a current need from the cell of 357 mA to do this work. From either the lifetime graph of this current or power (3rd and 4th graphs) we can expect something a bit over two hours.
Back to the start. In general, the less refined your circuit is, and the higher the 'jump ratio' in output over input voltage, the lower on the scale you're going to be. 'Quick and dirty' large boost circuits are sure to be closer to 50 than 90%.
FWIW, this last point (efficiency penalty for big boosts), as well as higher parts cost, is the reason traditionally this is done with the LEDs in parallel.
Doug Owen