Can I measure a journey time < 100 years on a 100 light year voyage? So, I'm traveling to another star 100 light years away in my spaceship. This ship has a solar sail pushed by a laser beamed from my home star system, so can achieve a velocity close to c. It's also got a robust parachute to slow down with.
I understand that if I measure light coming from my origin star, it will always still seem to be streaming past me at light speed (but will red shift as my speed increases). Light coming from my destination star also travels past me at light speed, and will become increasingly blue shifted as I gain speed.
I also understand that an observer checking on my speed at my origin or destination will always find it to be less than c.
However, will I perceive that in terms of the time it apparently takes me to reach my destination, my speed was greater than c? In other words, will it seem to take less than 100 years to reach the destination? 10 years on my watch, say. Or 1 year. Or a week?
Ie, as far as I'm concerned, while light keeps zipping past me at light speed, do I continue to accelerate unabated to an arbitrary apparent speed?
If not, how do I notice my continued acceleration being prevented?
 A: The following assumes that the distance to the star (100 light year) was measured before you got in the spaceship and started moving.
When moving close to $c$ in your frame of reference space around you will be contracted relative to what someone on earth will measure.  Thus, your 100 light year journey will actually be shorter in your frame of reference, and will thus take less than 100 years for you to make it to the star.
If you are ever able to report to someone back on earth that the journey took you less than 100 years in your frame of reference they will agree with you, since from the reference frame of earth your spaceship and all its inhabitants underwent time dilation, the slowing down of time relative to another frame of reference.  Thus, you would both agree on the time the journey took in your frame of reference, but the person back on earth will say that according to their clocks in the earth's frame your journey took 100 years.
In summation, you will not conclude that in your frame of reference you traveled faster than $c$ because while in transit, due to length contraction, the journey you traveled was actually shorter than 100 light years.
This is an answer to your bolded question, which is a different question than the one you posed at the end about whether or not your spaceship will accelerate forever.
The answer to that question is, as you continue to accelerate away from your origin star the laser propelling your spaceship will be redshifted (not blueshifted as you mentioned in your question) so that you will have less energy per photon than you had at the beginning of your journey (before the photons were redshifted).  Additionally, as you get further away from your origin, less and less photons from the laser will reach you; although lasers produce very focused beams, there are quantum limits to how focused these beams can be, so far away from the laser the spread in the beam will cause less and less photons to actually reach you.  These two effects combined will cause the energy you can extract from the laser to accelerate your ship to diminish as you get further away until effectively you can imagine that the acceleration the laser provides your ship vanishes.
