How many LEDs can you chain in series?
It's in series, not serial.

Assuming the LEDs are the same, those "further down the chain" aren't going to be any dimmer.  Since the LEDs are in series, they all have the same current through them, and will therefore have the same brightness.

To answer the question otherwise, do the math.  You have a 12 V supply and each LED might need up to 3.5 V.  (12 V)/(3.5 V) = 3.4 LEDs max.  Since you can only have whole LEDs, the answer is 3.



To calculate the current limiting resistor value:  You know each LED will drop at least 3.2 V, which means 3 of them will drop 9.6 V.  That leaves 2.4 V across the resistor.  You want the current to be 20 mA, so the optimum resistance for max brightness is (2.4 V)/(20 mA) = 120 &Omega;.