How many LEDs can you chain in series?

1. It's in series, not serial.
2. 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.
3. 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 Ω.

Each one you add will make the chain dimmer. You could run 3 (using a lower value resistor) in series.

If the forward voltage of all 3 was 3.5V you'd have a current of 1.5/R, if the forward voltage of all 3 was 3.2V you'd have a current of 2.4/R or about 60% more. You do have to be be careful of this if you're running the LEDs close to their maximum capability.

Edit: If you need to run 21, say, you can run 7 parallel strings of 3 LEDs in series, so you need 7 resistors total. Each resistor will be about 1.95V/ILED where ILED is the current you want the LEDs to pass. 470R would give you about 4mA, so you could run as many as 600 LEDs from a 10W supply (but 4mA may not be as bright as you'd like).

The answer should be the source voltage divided by the voltage drop of 1 LED. So if your source is 12v, and your LEDs have a voltage drop of 3.5, 12/3.5v, gives you 3 LEDs. If the the forward voltage drop of the LEDs was exactly 3 volts, you'd be able to drive 4 of them, and would not need a current limiting resistor.

The resistance of the current limiting resistor will vary with how many LEDs you're driving in series.

If you have 3 3.3 volt LEDs, you need to figure out the amount of leftover voltage to calculate the current limiting resistor. Thats 12-(3.3*3), or 2.1 volts. Use 2.1 volts in your current calculation. 2.1/.02= 105 ohms. You want the next larger E24 resistor value, or 110 ohms. If you assume your LEDs have a 3.2 volt drop, you'd use (12-(3.2*3))/.02 to calculate your current limiting resistor. That gives 120, which is an exact match. You'd use a 120 ohm resistor.

Edit: If you want to run 20 LEDs, arrange them in groups of 3 series LEDs, each with a 120 ohm resistor, and all the 3 LED chains wired in parallel. Like this:

^{simulate this circuit – Schematic created using CircuitLab}

P.S. A 4.7k resistor would probably prevent even 1 LED from lighting. It would only allow about 1.8 mA to flow through a single LED from a 12 volt supply, and with 3 in series you'd only get about 4.4 microamps through each LED.