How long did inflation take to happen? If inflation happened, it ended about 13.7 billion years ago. But how long was inflation going on before that? Was it just a short burst of time so that it's really okay to say that the age of the universe is 13.7 billion years, or was it going on for a long time before it ended (maybe forever?)?
 A: As the introduction to the wikipedia article states, inflation is generally considered to have occured $10^{-36}$ to $10^{-32}$ seconds after the big bang. This is extremely extremely tiny period of time. It is okay to say that the universe is probably approximately 13.7 billions years old.
A: The answer to this question is very hard to give without a theory of the start of inflation. There is no firm consensus, because the details of inflation are shrouded in mystery.
Standard new inflation
The way in which you find the approximate time which inflation happens is to integrate the slow-roll conditions. Slow roll means that the inflaton field is sliding down the potential well like a ball immersed in honey sliding down an incline, and you can tell how it slid from the shape of the potential. So you go back until you get to some Planck-scale cosmological constant or field value.
The Wikipedia estimates I presume, are for standard inflation models with a scalar and a polynmially rising potential. Then the time-scale for inflation is on the order of the doubling time at the inflation scale determined by CMB observations.
Stringy inflation
In the string-theory plausible brane inflation, some branes slide toward each other, then annihilate. In this situation, the situation where the branes are far apart is the initial condition, and it doesn't necessarily terminate on a singular big bang, but in another regime. To figure out what happens at the big-bang in these types of theories, you need to know the exact string solution and the exact brane configuration before annihilation.
Universe nucleation
There are other situations where you start the universe in an inflating state just as an initial condition. This is the Hartle-Hawking idea--- the universe tunnels into existence as a small sphere (a small inflating de-Sitter space), and there is no time near the beginning, so  it is impossible to ask how long inflation lasted, because the concept of time breaks down.
Eternal inflation
In the theory of eternal inflation, the universe never stops inflating, creating new bubbles all the time. This type of theory doesn't have a backward singularity, because it replenishes the regions where inflation is going on by using the notion of an enormouse volume of space outside the cosmological horizon is. In this theory, inflation never started.
In my opinion, this type of theory can be excluded by the general principles of logical positivism, which forbid us from speaking about things we cannot observe, like an eternally inflating region permanently outside our horizon. The proponents of eternal inflation hope that some sort of statistics on the eternal volume will give observable consequences to observations in our patch.
But in eternal inflation, the "likeliest" volumes of the universe are inflating for as long as possible. This gives rise to paradoxical consequences, noted by Guth. The paradox is that the amount of universe which inflates for 1 more second is vastly larger by volume than a region where inflation stopped one second ago. So imposing the final condition that we are here, we conclude that, if we are in a typical space-time volume, then we should have emerged as quickly as possible, to allow the inflation to go on as long as possible. This is absurd, because it requires improbable conspiracies in our history, like having the solar system and planets emerge as quickly as possible, etc, etc, with each second of time saved weighted by a factor of 10 to the bazillion.
These types of paradoxes can be avoided by restricting attention to one causal patch in cosmology. This point of view has been recently explicitly stated by Susskind, and it has also been advocated by Polchinski, but I think it should really be classified as some old holographic principle folklore.
