If I connect 10 3mm or 5mm blue LEDs to 3 AA battery power source, do I really need resistors?
AA batteries are a bad choice for powering LEDs due to the 0.8V cutoff voltage.  As the battery discharges from 1.5V to 0.8V the current changes significantly.



You should never connect LEDs to a battery without a resistor or a current limiting device like a constant current regulator (CCR).  LEDs have a forward voltage which will be different from the battery voltage.  Without another device (resistor or CCR) the LED will be forced to operate at the battery voltage rather than its Vf.

When using AA batteries and resistor you must look at how the resistor value works for both 1.5V and 0.8V.  

So 3 AA in series
@ 1.5V = 4.5V
@ 0.8V = 2.4V

You should connect all 10 separately each with its own resistor.

You could try all 10 wired in series but there is more likely to be an LED that cannot tolerate the stress.   I will connect 3 LEDs in series if their Vf are fairly close i.e. binned LEDs.

Blue and green LEDs have the highest Vf of all LEDs. Typically with a 2.9 to 3.3 Vf.  

I would recommend a very bright blue LED like Cree C503B-BAS-CY0C0461 with an intensity of 19,380 mcd.

You need the actual Vf when selecting the resistor(s).  The published values will differ too much and using AA batteries does not leave much wiggle room. 

Using 3.2V of the Cree LED:

At 4.5V the 1% resistor needed for 20 mA is 66.5Ω.
At 3.4V the minimum this LED will work properly the resistor would be 10Ω for 20 mA, 

You need to use the 66.5Ωbefcause if you use the 10Ω the LED will burn up at 4.5V.

See the problem with AA batteries? 

This is why you would need a very bright blue LED so you can see it well at 3 mA which is where it would be operating at 3.4V. 

When using a battery and resistor the best bet is to use a Li-ion battery which operates from 4V down to 3V. You would need a way to limit the battery voltage and not let the battery discharge below 3V.  



If you want to use AA batteries you should use a boost regulator. 

Connect the batteries in series and use a boost voltage regulator with a 3.3V output. 

The other option is to use a multiple output constant current regulator.  



You could use two MIC48125 output step down CCRs, powered with an LI-ion 18650 battery.

This appears to be the best low cost solution available. 

It has a constant current output, up to 20 mA, on each output which is set with a single resistor.  Input voltage can vary from 5.5 V down to almost the Vf of the LEDs.  Perfect for an Li-ion battery.