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So, you'll need three \$V_{BE}\$ (plus a little extra needed for \$R_2\$ and, if necessary, \$R_3\$ [which can be shorted]) to make this thing start to work. But it will have some independence of LED current vs \$V_{CC}\$, as well. Set of \$R_3\$ to have a multiplier effect if you don't like wasting too much excess current (over the LED's current.) But be aware that this also increases the variation of current over variations of \$V_{CC}\$. Set \$R_1\$ so that it provides the appropriate amount of current on this side of the mirror. There's one \$V_{BE}\$ across it, so this is easy to do. Set \$R_2\$ so that there is always at least 10% of the current in \$R_1\$ in the collector of the NPN (at the minimum allowable \$V_{CC}\$ you decide on.)

So, you'll need three \$V_{BE}\$ (plus a little extra needed for \$R_2\$ and, if necessary, \$R_3\$ [which can be shorted]) to make this thing start to work. But it will have some independence of LED current vs \$V_{CC}\$, as well. Set \$R_3\$ to have a multiplier effect if you don't like wasting too much excess current (over the LED's current.) But be aware that this also increases the variation of current over variations of \$V_{CC}\$. Set \$R_1\$ so that it provides the appropriate amount of current on this side of the mirror. There's one \$V_{BE}\$ across it, so this is easy to do. Set \$R_2\$ so that there is always at least 10% of the current in \$R_1\$ in the collector of the NPN (at the minimum allowable \$V_{CC}\$ you decide on.)

Okay. So that was fun. I probably should not have added the above circuit due to the level of disconnect between my reasons for adding it and my ability to communicate that reason sufficiently well. So, with the added desire of some measure of independence of \$V_{CC}\$ as well as the ability to work with low overhead as well, I offer the following thought. It crossed my mind to add during the discussion, but at the time my shift was over and I had to head off to bed. So please enjoy the following so that we can have still more interesting discussion.

^{simulate this circuit}

So, you'll need three \$V_{BE}\$ (plus a little extra needed for \$R_2\$ and, if necessary, \$R_3\$ [which can be shorted]) to make this thing start to work. But it will have some independence of LED current vs \$V_{CC}\$, as well. Set of \$R_3\$ to have a multiplier effect if you don't like wasting too much excess current (over the LED's current.) But be aware that this also increases the variation of current over variations of \$V_{CC}\$. Set \$R_1\$ so that it provides the appropriate amount of current on this side of the mirror. There's one \$V_{BE}\$ across it, so this is easy to do. Set \$R_2\$ so that there is always at least 10% of the current in \$R_1\$ in the collector of the NPN (at the minimum allowable \$V_{CC}\$ you decide on.)

Again, this is just me going off on a fun jag and responding to the discussion. Use matched BJTs as appropriate. There are still other approaches, such as the Wyatt which is dead flat over a wide temperature range and \$V_{CC}\$ range and if I worked on it might operate from almost as low as \$2.5\:\textrm{V}\$ and uses three BJTs (plus a mirror I might add then.) But then I'd have to explain why it achieves that and depends upon the 3300 ppm per degree change in resistance found in copper wire and metal film resistors as part of its temperature independence. There is an article on that from the 1990's, somewhere.

It sounds as though you are over-thinking the problem.

1. For example, a white LED may exhibit perhaps \$\frac{1}{4}\:\textrm{V}\$ change in its voltage for a factor of \$\times\: 2\$ in the current through it. But two different white LEDs from the same batch might exhibit as much variation, just one to another.
2. Also, LEDs are pretty tough and are often used in pulsed (multiplexed) modes where the peak current is much higher than the average. And they can usually handle it just fine.
3. Finally, human awareness of brightness is fortunately logarithmic. So a change in current in the LED by a factor of \$\times\:2\$ means a change in perception of brightness change that is barely perceptible (unless the LED is flickered intentionally with the two different currents to make it easier to perceive.)

So, all in all, the exact level of current usually isn't so important when an LED is used as an indicator light. And the voltage across an LED doesn't vary all that much, anyway.

The main thing is to make sure that there is sufficient voltage overhead to actually operate the LED consistently in a design and that the method of regulating the current is sufficient for the need (whatever that may mean) and doesn't cost too much (...) and doesn't take up too much space (...) and doesn't heat up surrounding things it shouldn't (...) and doesn't drain the battery more than necessary (...) and otherwise doesn't interfere with other design specifications (whatever those may be.)

In short, there are usually way too many other concerns to be worrying over.

[If the LED is used as one of three RGB LEDs, with the intent to use it as an LED pixel in a large external display, then it may be very important (or not so, depending on the requirements) that the currents are carefully calibrated in each of the individual LEDs in order to ensure actual design criteria such as "white balance" can be met. (Besides any LED "binning" that may have been done prior to assembly into an RGB pixel.)]

You present a problem, regarding LED current, where the problem uses a low-overhead voltage and exaggerates the problem by having LED voltages vary quite a bit (which, I suppose, might happen.) There is a modest solution to such cases, though I can't say anyone would care to bother putting three BJTs and a resistor to the problem. But let's say you actually have a design goal of "low overhead" control and consistent current control regardless of LED voltage variation. In such a case probably the cheapest method is to use a current mirror, as follows:

^{simulate this circuit – Schematic created using CircuitLab}

Even moving into saturation, \$Q_1\$ will still deliver a fairly consistent current (within 1% or so) to the LED and it will do so with only a few hundred millivolts of overhead. (Shorting \$Q_3\$ and removing its grounded collector would mean 10% variation moving into saturation, which still isn't horrible.)

It sounds as though you are over-thinking the problem.

1. For example, a white LED may exhibit perhaps \$\frac{1}{4}\:\textrm{V}\$ change in its voltage for a factor of \$\times\: 2\$ in the current through it. But two different white LEDs from the same batch might exhibit as much variation, just one to another.
2. Also, LEDs are pretty tough and are often used in pulsed (multiplexed) modes where the peak current is much higher than the average. And they can usually handle it just fine.
3. Finally, human awareness of brightness is fortunately logarithmic. So a change in current in the LED by a factor of \$\times\:2\$ means a change in perception of brightness change that is barely perceptible (unless the LED is flickered intentionally with the two different currents to make it easier to perceive.)

So, all in all, the exact level of current usually isn't so important when an LED is used as an indicator light. And the voltage across an LED doesn't vary all that much, anyway.

The main thing is to make sure that there is sufficient voltage overhead to actually operate the LED consistently in a design and that the method of regulating the current is sufficient for the need (whatever that may mean) and doesn't cost too much (...) and doesn't take up too much space (...) and doesn't heat up surrounding things it shouldn't (...) and doesn't drain the battery more than necessary (...) and otherwise doesn't interfere with other design specifications (whatever those may be.)

In short, there are usually way too many other concerns to be worrying over.

[If the LED is used as one of three RGB LEDs, with the intent to use it as an LED pixel in a large external display, then it may be very important (or not so, depending on the requirements) that the currents are carefully calibrated in each of the individual LEDs in order to ensure actual design criteria such as "white balance" can be met. (Besides any LED "binning" that may have been done prior to assembly into an RGB pixel.)]

You present a problem, regarding LED current, where the problem uses a low-overhead voltage and exaggerates the problem by having LED voltages vary quite a bit (which, I suppose, might happen.) There is a modest solution to such cases, though I can't say anyone would care to bother putting three BJTs and a resistor to the problem. But let's say you actually have a design goal of "low overhead" control and consistent current control regardless of LED voltage variation. In such a case probably the cheapest method is to use a current mirror, as follows:

^{simulate this circuit – Schematic created using CircuitLab}

Even moving into saturation, \$Q_1\$ will still deliver a fairly consistent current (within 1% or so) to the LED and it will do so with only a few hundred millivolts of overhead. (Shorting \$Q_3\$ and removing its grounded collector would mean 10% variation moving into saturation, which still isn't horrible.)

With low overhead situations, a resistor makes a very poor current regulator. That's just how it is. So you either live with it, or not, depending on the circumstances.