In Germany, at the Bitburg Aiprort , the former Bitburg Air Base , there is such a possibility.
Well, they don't have MiG's, but you can have a flight on an L-39 Albatros , a trainer aircraft.
There is a website in German .
If want you to have more information, you can try to contact the guys.
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Getting a ride in an ex-military jet is totally doable; pilots love to fly, and it's even better when someone else is footing the bill.
Ex-military aircraft in private ownership are often called "warbirds", and searching for that term will provide many leads.
Here's one:
http://www.warbirdalley.com/fly.htm Take a ride in a warbird!
Here are links to various organizations which actively operate warbird aircraft, including pay-per-flight operations and museums.
We've also listed a few miscellaneous other ways to get some hands-on flight experience with these exciting airplanes.
Or: http://www.flyfighterjet.com/
You can fly a jet fighter such as the Supersonic MiG-29 Fulcrum interceptor, the Aero L-39 Albatros or the Hawker Hunter fighter-bomber aircraft.
We offer fighter jet rides from various airfields all over the World.
This is no simulation: Here you can book a real jet fighter ride in a military aircraft If you're serious, you could even troll the FAA aircraft registration database for exotic planes, and contact the owners directly (or use your findings for more targetted googling).
Another fruitful technique would be to simply google for particular aircraft types.
For example, the L-39 is a Czechloslovak jet trainer of which there are many in private hands.
Googling reveals the existence of the L-39 Enthusiasts group, which could certainly direct you to a pilot happy to give you a ride.
One site quotes $3000 for a 45 minute ride .
Finally, aviation is sometimes a bit old-fashioned, and the best way to find what you want is often to simply show up at the airport and ask around.
Someone will know someone and soon you will find your way into the cockpit of a warbird.
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adrenalin.com.au has some offers for Australia, and you have the choice between a L-39 fighter jet and a A37B ex-military jet.
Prices between AUD 1,200 (15mins flight) and AUD 2,700 (35mins flight) at the moment.
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German event organizer Jochen Schweizer offers flying a MiG 29 in Nizhniy Novgorod for 14.5k EUR.
bestrussiantour.com has an offer for flying a MiG 29 and/or
Mig 31 at the same place, with a price list for download.
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Ken Thompson wrote an article about flying a MiG29 in Moscow.
He booked the flight via a Miami based organization called Fly-With-Us.
The article is a fun read, though possibly a bit dated: I discovered it many years ago.
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There is a company in Florida that uses L39 Albatross planes and others for NASA type astronaut training.
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I remember that there was a possibility that one could take a ride in a military MiG.  I think at one point the cost was $15k.  Are there any other places that would allow a person to take such a ride?
Riding in a military jet
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There are options in several countries, including Bitburg Airport in Germany, with adrenalin.com.au in Australia, you have the choice between a L-39 fighter jet and a  A37B  ex-military jet, with a company in Florida  that uses L39 Albatross planes and others for NASA-type astronaut training or via the event organizer Jochen Schweizer, which offers a flight in a MiG 29  in Nizhniy Novgorod for 14.5k EUR. Aside from this, you could simply Google particular aircraft types or ask around at the airport itself. Pilots love to fly, and it's even better when someone else is footing the bill. Google is a good way of finding out about flights in military aircraft as are othe rwebsites such as adrenalin.com.au. Many companies offer flights, however, it may be possible to find something by just turning up at the airport and asking.
It is possible to get a flight in a military aircraft. Several websites offer exactly that, for example adrenalin.com.au. There are companies specialising in this activity in Germany and Florida, USA. Using Google or going to the airport are other ways of finding out more.