I am sitting on a plane from Rome to Philadelphia, marveling at how quickly we can move around the globe. There is a 6 hour time-zone difference between Rome and Philadelphia. My flight took off at 11am in Rome and was set to arrive 9 hours and 45 minutes later at 3:45pm in Philadelphia. This got me thinking, what would your daily schedule look like if you continuously flew west?
My calculations are based on gross approximations from the information given in the seatback display in front of me. For example, I am assuming that you could fly continuously forever, and I am assuming that the rate at which my flight is changing time-zones would be representative. We will also not take seasonal changes into account and assume a 12 hour day/night cycle.
Stupid simple calculations
First, I am going to calculate the effective rate at which time is progressing for me in the air (relative to the day/night cycles for someone fixed on the ground).
effective.rate <- (15.75-11.00)/9.75 (effective.rate*60)
##  29.23077
That is, for every hour that passes on the ground (with respect to day/night cycles), I experience only 29.23 minutes of progression in the day/night cycle.
Since there are no appreciable relativistic effects occuring at 858 kilometers/hour (our speed according to the display), we can say that I experience the passage of 1 hour identically to the experience of someone remaining in the same location on the ground. In that way, I do not feel that 1 hour feels longer or shorter while in flight. Now I will calculate the number of hours it would take to experience a full day/night cycle when traveling at this rate.
##  49.26316
Therefore the passage of one full day/night cycle would take about 49.26 hours or that day/night would each last about \(24.63\) hours.
Thats a pretty long day.
A little farther
I have long said that my internal clock is acctually a 26 hour schedule. Effectively, if left to my own devices, I tend to go to bed about 2 hours later every night and wake up 2 hours later every morning (doing a full cycle every 12 days). So what speed must I be traveling west at to achieve this desired 26 hour day?
I will start by calculating the time it takes to cross 1 time-zone and the resulting estimate for the average number of kilometers per time-zone during my trip.
current.speed <- 858 # kilometers per hour (km.per.tz <- (9.75/6)*current.speed) #(hours in flight/time zone difference)*current speed
##  1394.25
desired.cycle.length <- 26 (1-24/desired.cycle.length)*km.per.tz
##  107.25
Therefore, if I traveled west at about 107.25 km/h my day would appear to be 26 hours long.
These calculations are not completely correct but I am guessing I am within about 25% of the correct result based on this physics stackexchange post. Under certain conditions they calculate that at the equator traveling at 1672.2 km/h would allow you to track the sunset. I find that you would have to travel at 1394.25 which is 0.83% of their answer.
My approximation would likely be much better if I considered that my flight was not at the equator and that my speed and direction (velocity) was not constant throughout the entirety of the flight.
Either way I find this an interesting thought experiment.