Electronic – Need assistance with current limiting resistor for LED

Your calculation is correct. linear1 rounds up to the next E12 value, which happens to be 100\$\Omega\$. The nearest E12 value would have been 82\$\Omega\$, and that would still be safe, because, even if the current will be higher, the difference will be small, within the 10% tolerance of the E12 series.

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Purists may say I'm cutting corners here. Russell has a long answer about iterating the solution, and others whine (hey, no offense!) about rounding up being more safe. My answer is meant to be pragmatic; no professional design engineer can afford to spend 15 minutes to calculate the resistor for a classical color LED. If you stay well below the maximum allowed current you'll have enough headroom to allow some rounding, and the rounded value won't be noticeable in brightness. For most LEDs perceived brightness doesn't increase much above a value of typically 20mA, anyway.

Have you tested the two alkaline batteries? I'd bet they are slightly above 1.5V each, bringing your total voltage up to the LED forward voltage. However, when the LED starts conducting, its low impedance pulls the battery voltage down slightly, and the battery itself acts as the current limiting resistor.

As to the second part of your question, I can't say whether its always better to add the resistor and use more cells. If you're making an LED throwie, then it definitely isn't. If you want a consistant runtime that doesn't rely on hitting a bullseye between battery voltage and current draw, then probably.