Accelerating a person's perceived sense of time is a fairly common subject in science fiction, and to keep a long story short, I was wondering just how far you could stretch time out when you're working with an organic brain rather than a digitally uploaded mind. Is the limit something like experiencing a subjective ten seconds for every one second relative to the rest of the world, or could you take it further? Would a hundred or a thousand times normal speed still be in the realm of possibility? What about a million times?
If you've ever experienced this stretched perception of time, you may have noticed that while your perception speeds up (that is, you see the world in slow motion), your reaction time doesn't speed up by anything like the same ratio.
There are two factors at work here. One is the neuron transmission time: that is, the time it takes sensory input to get from say the eyes to the visual cortex, then for a response to travel from the brain down the motor neurons to muscle cells. The other is the brain's calculation speed, which (since the brain is highly parallel) can be increased by training, moving some things from conscious thought to reflex action.
In the current state of knowledge, there really doesn't seem to be a lot you can do to speed up the fundamental "clock rate", since it depends on the speed of nerve conduction, which in turn depends on ion channels & neurotransmitter releases. Something like adrenaline can (I think) potentiate some of this, but only within limits.
Sticking with known science, I'd make a semi-educated guess, and say you could probably speed up perception by 5:1, reaction time by 2:1.
A brain "calculation speed" or "clock rate" is pretty much limited by the related metabolic rate.
Assuming there is a linear relation between the two rates, if you want to speed up the clock by a factor of 10, you need to do the same with the metabolism, which is already optimized and limited by chemical reactions at body temperatures.
And while you can increase the fan speed of an overclocked CPU, enzymes in biological bodies stop working out of their design temperature.
Alternate Biology OR the Inorganic Brain:
Short of actually altering the flow of time around a brain, the perception of time could be sped up by increasing the speed of the brain. Biology only allows for information exchange at a certain rate (the movement of molecules), so a non-chemical brain becomes important. This does not rule out biology if you have information transfer in the brain by something like entangled particles. Alien conspiracy theorists talk about "monoatomic gold" speeding up thoughts. Perhaps this could be a means of transmitting data at FTL speed from one neuron to another. An alien with some sort of solid-state(crystal, etc.) brain could approach computer speeds. Or assuming a hyperspace reality, a character could have his cognitive functions in another reality where physics ran differently and translated his thoughts to our universe.
Let's be clear that physical duration is a constant vector. It is bound by the laws of relativity to progress at the same speed. We are limited by this fact. Now, the only way to make time seem slower is to cram more processing power into each segment of time. If you have a fight-or-flight response, your body will trigger a short burst of what I'll call "brain overclock," where we become hyper-aware and ready to react more quickly to external stimuli. I would venture to distinguish reaction from deliberate action, which happens much more slowly, and requires much more processing power to perform.
Also, we are limited by physics. Even if you could somehow perceive time "more slowly" than the rest of the world, you still couldn't run the 40m dash any faster than you could in "normal time".
I think there's something wrong with trying to give a ratio of relative duration, or perception of time, because all time is linear. It's the speed at which our brain calculates that makes time seem to progress more slowly, and even that is unsustainable as you can see from the other answers.
Perhaps a shot of epinephrine, or some crazy amphetamine salt concoction could give you a temporary boost in brainpower, but can a person subjected to that honestly say that, for instance, time dilation is 2:1 in their favor? I don't think its quantifiable in that way.