Can an observer watch billions of years of history in a few hours? Say a spaceship leaves Earth today, travels for x years at 0.9999999c, and then decelerates and stops (back to Earth's frame of reference, but far away). Almost everything else in the universe will now be much older.
If deceleration is not discrete, but continuous, it means in y hours of deceleration, potentially millions of years of space history would be observed by the spaceship. Is this correct? What am I missing?
 A: Judging by your comment on Claudio's answer, you have understood everything the wrong way around. When two ships are approaching each other, each will 'see' time on the other speeded up, owing to the Doppler effect. (I put the word 'see' in quotes because in reality you would never see anything in the everyday sense of the word if you were moving at close to the speed of light.) As the relative speed of the spaceships decreases, the magnitude of the Doppler effect will reduce, so that when the two ships are stationary relative to each other they will eventually see time on the other running at a normal rate.
I have included the word 'eventually' for an important reason, which is that if one ship decelerates, they will immediately see a reduction in the Doppler effect that applies to light coming from the other ship, but there will be a time lag before the other ship will see the same reduction in relation to the light it receives.
A: It is not only during the deceleration time that one ship sees the other in 'fast forward mode'. Because they are approaching, it happens during all the journey.
What happens during the deceleration time is that this mode gradually turns to 'play'.
In case 1, it happens equally for both.
In case 2, it is possible that STATIC sees FAST even at slow motion during that time, in spite of being approaching, if the deceleration is big enough.
