The 20mA rating of your LEDs is probably the recommended maximum current. Most LEDs will produce light at much lower currents - I find 5 mA is sufficient for indicator light use.
Try the LEDs with different resistors to see what current is actually required to produce sufficient light for your application (more current = brighter light).
Also, check the maximum current the Pi GPIO pins can handle - I have a feeling it is less than 20 mA, but can't check at the moment.
Background:
I have designed a number of LED lighting products which are manufactured in China.
I have several cylindrical LED flashlights that have a large number of LEDs in them
... Are there ultra-bright LEDs that you can drive directly off of 4.5 volts without a current limiting resistor? Or are there special purpose ultra bright white LEDs made for 4.5 volt supply that have internal current limiting resistors?
No and no, unfortunately.
Many LED lights are constructed as you describe, with multiple white LEDs wired in parallel and connected essentially directly across the battery.
They are junk.
They are not "designed".
They build them this way "because they can" and they work well enough to be able to sell them.
When supplied with 4.5V + the LEDs are driven well above their maximum design rating and their lifetimes are greatly shortened. The LEDs used are typically low lifetime low cost devices.
Follow-up question: Does anybody know if the 12 volt LED bulbs that are in landscape lights have a voltage regulator in them?
The 12 volt LED strips usually use 3 LED die in series plus a series resistor.
Turn on / turn off time is liable to be sub `1 microsecond if capacitors are not used downstream of the switch.
Current is set to be "about right" at 12 Volts so will vary substantially if used in an automotive context where several volts of variation occurs. Many strips use individual LEDs but some use 3 die per package LEDs with all 3 independent die wired in series. It is possible but not certain that strips with individual LEDs will run somewhat cooler due to a lower concentration of Energy per package.
Lifetime of these LEDs may be better as the series resistor means that they are somewhat more properly driven. I have seen very substantial variations in output of similarly appearing strips. The brightness bears no obvious relationship to LED specifications and a brighter strip may simply reflect a manufacturers 'marketing decision'. You can get a range of LEDs per metre but current drain and number of LEDs are not directly related.
White LEDs are typically have a voltage drop in the 3.0 - 3.5V range at rated current.
Current increase tends to be exponential with voltage and at 4.5V almost any LED would self destruct almost instantly. The "saving grace" (if it can be called that) is that the combination of small batteries and many LEDs means that the batteries are unable to produce more than 'vastly too much' current when the batteries are new. Any light constructed in this manner demonstrates a total lack of concern and/or knowledge by the manufacturer.
Adding even a single common series resistor makes a substantial improvement in voltage/current profile and a resistor per LED would greatly assist current balancing between LEDs.
Added May 2016
Harper commented:
OP is asking about LED bulbs, not strips. Those are commonly made as screw-in replacements for incandescents. Some have a resistor, but many have a switching buck converter which will accept a range of voltages from 12-30V or higher. The LED series voltage is quite close to 12V actual, so if voltage drops much below 12V the buck converter will go to 100% duty cycle and simply pass the voltage through, causing the LEDs to dim rapidly.
My answer addressed LED strips as I noted, which the OP did not ask about, as Harper noted :-).
Harper's comments above are correct where applicable. I have not seen a bulb with a buck converter internally, but no doubt they exist. White LEDs have Vf typically in the range 2.8V - 3.5V. 2.8V is unusual and usually only seen in reasonably modern LEDs or ones operated well under full power. At 12V nominal, 4 LEDs have 12/4 = 3V each available. Allowing a small voltage drop in connectors and wiring 4 LEDs with Vf of 2.8V to 2.9V would be able to be operated at full power. In real world situations with Vin able to be somewhat below to substantially above 12V, 4 LEDs in series will often work but 3 x LEDs in series plus a series resistor is 'safer'. Bulbs may not match strips in configuration, but all 12V LED strips that I have seen use 3 LEDs in series plus a resistor.
Best Answer
I would say you want to get a handful of the LEDs and test them to destruction.
You will need to get or build a variable constant current source. This can be made with an LM317 or similar adjustable regulator - google for LM317 constant current. The current resistor should be a wire wound low ohmage rheostat ideally to take the current, plus a tiny (though high wattage) zero-stop resistor.
Power the LED with the constant current source set to its minimum and monitor the current value with a DMM.
Slowly increase the current until the LED starts to illuminate. Measure the voltage across the LED for your \$V_F\$. Slowly increase the current more and more, noting down the current value with each adjustment.
When the LED burns out the last current reading you took will be the absolute maximum current for the LED. Do not, I repeat do not run the LED at that current, as you are right on the cusp of burning it out.
Get a second LED and replace the blown one. Now adjust the current to just below where it burned out before, and this time slowly reduce the current down until you see the LED visibly reduce in brightness. This is the current that you don't really want to drop below as the LED won't be properly bright.
Now pick a value roughly half way between the two and calculate a resistor (or better yet a constant current source or sink) accordingly.