Well, let's analyze the circuit. We know that the power required in a DC circuit is:
P = Vsrc * Iout
We know that
I = (Vout - Vled)/R
and the power delivered to the LEDs is all that matters, so we want to maximize
Pr = (Vout - Vled) * I = (Vout - Vled)^2/R
Pled = Vled * I = Vled * (Vout - Vled)/R
Clearly, we want to minimize Pr and maximize Pled. We can do this without decreasing the current by reducing R and making Vled close to Vsrc.
This is accomplished by putting the LEDs in series.
However, your battery (isn't the 6LR61 a 9V battery?) will go from some nominal voltage (ex 9V) to a lower voltage - 9Vs are spec'd to be dead at 4.8V. This means that a passive solution will go dim while there's still charge left in the battery. For your original schematic, that might mean that you'd end up below the minimum current to turn the LED on, or for the series version, the voltage might go below the diode forward voltage.
A simple way to extract more brightness with the same power is to pulse the LEDs - Human eyes percieve blinking light to be brighter than continuous light, even if the average power is the same. A 555 timer or other oscillator/switch combination will be able to do this, no microcontroller required. Try playing with the duty cycle and frequency of your LEDs to see where it looks the brightest - You may be surprised!
Also, a switching power supply can increase the efficiency of your regulation circuit to 80, 90, or even 95%. However, that will drive up the cost and complexity of the design, and may not be necessary.
Any LED multiplexing solution will inherently reduce intensity of the LEDs pretty much linearly by the factor of the multiplexing duty cycle. This change in intensity is not identical to the change in intensity observed when current through the LED is reduced by changing series resistance: The resistance value is not directly the determining factor, the resultant current through the LED is, and that does not enjoy a linear relationship to resistance.
The human eye's perception of intensity of light, LED or otherwise, is far from linear: Intensity variations at low LED intensity, or for that matter in low ambient light conditions, are much more sharply perceived, than a similar change in intensity at a brighter point in the intensity curve, or under brighter ambient light conditions. This explains why the change in resistors did not show a significant change in perceived intensity.
To answer the specific question about over-driving LEDs: Yes, that is a standard mechanism for countering loss of intensity in multiplexed displays. To counter a 8-way multiplexed LED display's intensity drop, i.e. 12.5% duty cycle per LED, hypothetically the LED should be driven at 8 times the constant current. However, that is rarely practical, so either compromises must be made, or risks of LED damage must be accepted.
How much a given LED can be safely overdriven by, is (a) possibly specified in the datasheet, but (b) more typically determined by the display designer through empirical testing, especially in hobby and low-cost applications, if the datasheet-specified overdrive current is not sufficient, and risk of LED failure is acceptable.
- For more context on LED overdriving limits for pulsed load, see this answer.
- For a more detailed description of LED intensity and human perception, see this answer.
- Another answer in the same vein, with pretty graphs too.
Best Answer
As you calculated, with 5V, you can only power 2 leds in series. For multiple leds, you have to place them in parallel strings of 2, each with its own resistor.
If you are using a computer ATX power supply, you also have 12V available. To save on resistors, you could power 5 + 100 ohm resistor in series with 12V instead of 5V. Then you can do multiple strings in parallel.
simulate this circuit – Schematic created using CircuitLab