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I have a 15 year old ReplayTV DVR. I got it used, I'm not sure when, but probably about a decade ago. It's turned on almost all the time, and it has a blue power LED on the front.

The LED FAQ says that many LEDs have a rated lifetime of 50,000 hours, which implies that it should only last about 5.5 years when used 24x7. Green Efficient Homes says that "an individual LED may well last 100,000 hours", so that bumps it up to 11 years.

These sites seem to be about LED light bulbs, not the individual LEDs that are used as indicators on electronic devices.

Why has this little guy been able to keep shining continuously for so long?

Properly designed, built and used, today's LEDs have incredibly long lives and the wearout mechanisms are not catastrophic in nature, so instead of using incandescent lamp MTBF statistics, a luminosity percentage (70%) is often used to define the lifetime- this doesn't mean that the LED burns out, it means that the light out put is only 70% of what it was when brand new. Because the change is slow, you normally don't notice it.

For white light applications this makes sense because after a period of time the light works but may be too dim to be useful in the application such as a flashlight or reading lamp.

FWIW incandescent lamps are prone to something similar in that the filament would slowly evaporate leaving an dark coating on the glass shell, dimming the light. They are able to compensate a little bit because the filament glows a bit brighter due to the increased resistance (until it breaks).

The rating is only a MTBF (mean time between failure)

LED MTBF

For any component or system, the MTBF is the mean time between
failures. The MTBF is the elapsed time which is predicted between
inherent failures of a component or system during operation.

The MTBF is a figure used in calculations for the reliability of items
of equipment. In order to be able to calculate the MTBF of the
equipment, it is necessary to know the MTBF of the individual
components, e.g. the LED MTBF in this case.

The failure rate for a component, and the MTBF are linked. MTBF can be
calculated as the inverse of the failure rate if it is assumed that
there is a constant failure rate, which is not unreasonable as a first
order assumption.

MTBF = Hours of operation / Number of failures

The MTBF figures are often quoted in the manufacturers data sheets.
However the MTBF can be considerably reduced by operating components
close to their rated limits. Hostile environments such as high
temperature and vibration also reduce the MTBF.

However when run within their limits, the LED lamps have a long
lifetime, and do not fail very often.

Source: https://www.radio-electronics.com/info/data/semicond/leds-light-emitting-diodes/lifespan-lifetime-expectancy-mtbf.php

Without a sample population and testing of that specific LED it will be impossible to tell what the lifetime of that LED is. the manufacturer can only guarantee through what they know about that LED (probably through running it through months worth of testing, measuring the degradation) and then extrapolating that over the lifetime. They then guarantee an MTBF figure and if you get a batch of their LED's on average, they will last more than x amount of hours. But they could last more, the MTBF figure is only a lower bound.

Estimations of life expectancy of electronic components is a tricky business. As one can imagine, manufacturer can't wait 50,000 hours (or something like 2,290,000,000 hours for a RS232 receiver, see page 7 of this essay) to collect statistics of device failures, market doesn't allow this. All estimations are done using "accelerated models".

The idea is to set a batch of devices under extreme conditions (voltage, current, temperature, environmental factors) until some devices fail in a reasonable test time (days or hours), assuming no shift in failure mechanisms. Then, using some theoretical models of failure mechanisms and other ASSUMPTIONS, the failure rate gets extrapolated to normal operating conditions. Usually the theoretical models are of exponential type, so small errors during accelerated testing can result in vast differences in estimations for tails of these exponential functions. So the models are usually on a conservative side, to avoid liabilities from catastrophic premature failures.

For example, if the accelerated model for a LED did include the air humidity and rust/oxidation factor, but your LED was always at room and dry air conditions, it can last much-much longer than the manufacturer's specified MTBF.

Those times are rated lifetimes and can be very conservative. So it might well be that most or (in case you are lucky) some of them can handle a factor times more, so 15 years sounds not unreasonable.