Is there a commercial service offered to general public that allows you to celebrate the new year several times? The idea is that you would celebrate it in a time-zone and than move to the next one arriving there before new year. Of course this is relatively easy to accomplish by yourself if you're near the border of a country and you just move to the next one in a different time zone. Things probably get a bit more complex to do it 3 or more times.
There are four ways you could celebrate New Year's multiple times, but the TL;DR is that no, there's no practical way to celebrate New Year's more than twice... although there are a few theoretical possibilities if you're willing to spend a lot of money on this!
The best way is to cross the International Date Line, so you get an entire day or more to celebrate the New Year again. This is covered pretty thoroughly in this question and its answers, but since there's only one IDL, you can only do this once.
Barring that, you could hop across a land border with a large time difference, eg. you gain 3.5 hours when you cross from western China (UTC+8) to Afghanistan (UTC+4.5), although in this particular example neither place is exactly party central. As far as I'm aware, due to the size of the average timezone (~1600 km at the equator), you can't pull this off more than once in a day either, although there are a few places where three time zones meet. The least inconveniently located (h/t Henning Makholm) appears to be the town of Iñapari, Peru, next to the punto tripartito where Peru, Brazil and Bolivia meet. Alternatively, you could also go all the way to the North or South Pole and just walk around the pole, but I'd argue that's cheating a bit.
However, another answer cleverly points out you can combine this with option #1 to score four or even five New Years. Ka-ching! Although you'll need to charter a jet and sort out some fairly hairy border permits.
The third option, which I suspect you had in mind, is flying westwards faster than the planet rotates, so you can party over and over again. Alas, the planet rotates at 1,675 km/h at the equator, meaning that all of today's commercial aircraft are too slow, since they top out at ~1,000 km/h. Even around halfway between the equator and the pole (say, North America and Europe), the surface rotation speed is still too fast at 1125 to 1450 km/h.
Now the late, lamented Concorde could have theoretically pulled this off, since it could fly at over ~2100 km/h, and some current military jets do reach these speeds (or more!). But even if you had one at your disposal you'd have to pretty much keep flying if you want to keep popping the champers, since takeoff or landing at a busy airport can easily take half an hour with approach paths and ATC clearance and taxiing and whatnot.
But there is still one travel option that travels far faster than the Concorde, namely the International Space Station, which zips around the Earth in approximately 90 minutes and thus lets you celebrate New Years approximately 24 hours * 60 minutes / 90 minutes = 16 rotations x 40 timezones = 640 times in a 24-hour period if you're so inclined, although you'd apparently hit "only" 16 exact midnight-on-New-Year's-Eve moments. However, you're looking at upwards of $20 million for the ride, and the vagaries of scheduling mean you'd need to be pretty lucky to spend New Year's in orbit.
The most realistic option might be visiting the village of Tripartito, right where Bolivia, Chile and Peru meet. The three countries are in different timezones in the southern summer (when Chile observes DST but Bolivia doesn't), and at Google Maps it looks like there are roads crossing the borders right at the triple point, so you can simply leave one timezone at about 0:30 and walk over to the next arriving at 23:30. This will give you three celebrations of the same new year.
If you're willing to use a somewhat less accessible triple point, it looks physically possible to do 5 New Years with a bit of ingenuity and a chartered jet.
Because Russia's time zones are offset a few hours from mean solar, there's a point on the Heilongjiang river where three time zones meet. You would need a boat, your own party supplies, and an understanding with the local border guards.
(The same three zones meet at the north end of the Russian/North Korean border, but the logistics of partying there are probably not favorable).
After the Chinese new year, you have a comfortable 16 hours to get yourself the 4000-or-so miles to somewhere on the Canada-Alaska border and celebrate two more new years. Again the border looks pretty desolate, so BYOPS.
Getting 6 New Years will be cutting it uncomfortably close, but it might be possible to have your first party in the far west of China (UTC+08), then fly your chartered G650 to Kirkenes in the far north of Norway (about 3925 km), where you have a helicopter ready to hop you over to the Russian border and celebrate new year just past the border booth in UTC+03.
Walk back through the border crossing and the helicopter flies to nearby Finland. This is just a bit too far to do by road in an hour according to Google Maps (and road conditions that far north are probably not the best at New Year's anyway), thus the helicopter. Fortunately Norway and Finland are both in Schengen, so there should be no further red tape before you have your third new year in Finland, UTC+02.
Helicopter back to Kirkenes and celebrate New Year in Norway's UTC+01. You'll then have plenty of time to fly about 5300 km to the Yukon-Alaska border and celebrate New Year in UTC-08 and UTC-09.
The main risk with this plan is weather conditions at the Russia/Finland/Norway triple point, in the polar night far above the Arctic Circle. Also, the China-Kirkenes leg is a big rushed; a bit of headwind here could ruin the timetable. (Perhaps land in Murmansk and afterwards fly the jet across the borders instead? Would save the helicopter too.)
The same three timezones meet at Belarus/Ukraine/Poland and Belarus/Lithuania/Poland, but there don't seem to be suitable airports close to those triple points.
Another alternative would be to replace Kirkenes by the Pakistan/Afghanistan/Iran triple point. There's an airport nearby in Zahedan, Iran, but there doesn't seem to be any official crossings of the border, so again some special arrangements with the local authorities have to be made. Also, the transpolar flight times will be iffy; you may need a gravel kit for your G650 so it can land at Beaver Creek rather than Northway.
Visit the Amundsen–Scott South Pole Station, and walk around the South Pole in 24 hours such that the South Pole is always between you and the sun.
Now, Amundsen-Scott normally operates under New Zealand Daylight Saving Time (Time in Antarctica). However, for the purpose of New Year celebrations, use Solar time instead. As you move around he pole in the described fashion, solar time will always be midnight, so you can celebrate New Year for 24 hours continuously.
Well, there is a little problem - you can't visit the station as a tourist. So you would have to apply for a job as researcher there.
A fairly simple way (unless I'm misunderstanding the question) to get three celebrations would be to go to Cameron Corner in Australia, on the border of New South Wales, South Australia, and Queensland:
You would have your first midnight in NSW (in Australian Eastern Daylight Time), then 30 mins later step into SA for another midnight (in Australian Central Time), then finish with your final midnight 30 mins after that by stepping into Queensland (in Australian Eastern Standard Time).
You could also get three celebrations at two other corners in Australia (SA-Qld-NT and WA-NT-SA), but I think Cameron Corner is the easiest to get to (there's a 4WD track that goes to it and a nearby corner store).
Here's a nice map showing Australian timezones during the Austral summer, when some states have daylight savings time:
A quick google search indicates there was an offer of doing this by private jet from Sydney Australia (not Nova Scotia) to Los Angeles California, if one happens to have 130,000 quids (sic) burning a hole in one's pocket.
The timing is rather tight for commercial flights to be successful from the other suggestions, and usually flights don't take off after midnight anyway because of noise concerns. The folks offering the expensive junket claimed there was no way to do it on commercial flights.
I do note that courier and air freight flights take off from some Chinese airports after midnight, so it might be possible for someone with the connections (perhaps a deadheading pilot) to hop aboard such a flight.
Of course, for those on a budget, one could simply celebrate New Years in Terra Haute IN (USA) and have your designed driver drive the 21 miles west to Marshall IL (pop. 3,933) and celebrate once more. Or pick some other places just across a time zone boundary..
I believe there was a visit to the North Pole via icebreaker a few years ago. Don't think it was available over New Year though. But if available standing near to the pole it would be trivial to walk round and catch every time zone. Just make sure your partying doesn't involve large quantities of alcohol.
2 minutes on google found this