… and I have an infinite supply of money and have bribed various government officials, so I have access to any technology that currently exists.
How long would it take to get to the other side of the planet, and by what means would that be?
Note: this is only using technology and infrastructure that currently exists.
45 minutes, by ballistic suborbital trajectory on an ICBM. Advantage: high-accuracy delivery (<100m CEP). Disadvantages: the landing is a bit bumpy; flight too short for a feature film; other in-flight services limited.
I doubt that you can beat ordinary air line scheduled service for the trip. Your bribes, and special projects are going to take longer than just buying a trip that will pass through the antipodes. You might need a charter for the part that just goes out into the ocean and throws you off into the air. A good travel agent could handle the desk work while you fly the first leg.
Probably a bit under two days, unless you get a really great set of connections.
Tris
Wouldn’t the g-forces generated during acceleration phase pretty much preclude arriving at your destination alive? Don’t ICBMs accelerate much faster than manned rockets?
Triskadecamus has it, that in many cases it’s only the last part of the journey that’s the problem. Expedia tells me I can fly from Norwich to Christchurch in 28 hours. All that’s left is driving to the airport, check-in time, customs, and chartering a ship to take me 600 miles into the ocean.
Well, by surface travel only, by boat, road and rail, but not air you can do it in about 40 days.
Start digging!
That’ll take longer than flying. silly! 
At the equator you can accelerate at 1g, decelerate at 1g, and make the trip in around 45 minutes. I don’t think that ICBMs follow that acceleration profile though.
I thought price was no object. Just fire up the exawatt laser.
Well, pay someone to invent the exawatt laser, first.
And how long’s the exawatt laser gonna take to build, eh? Eh? Eh?
The fastest currently available means I can think of would be to bribe or persuade military planes to do an in-flight refueling for non-stop flying. I’m not enough of an expert to know which are the fastest planes capable of the 12,000 mile trip, but it seems safe to say that you could do it in under 12 hours.
I may be talking out my buttocks here, but I know the US sells supersonic fighters to other countries. Might it be possible for you to negotiate some kind of special rental? Or even buy one yourself, sans weapons and top secret items?
Or if not, I know you can rent flights on Russian interceptors and fighters. I’ll bet you could do a sprecial rental for one of those, like the MiG-25 Foxbat.
Consider the first paragraph probably unjustifiably off-topic for GQ; but I think the second one is a valid suggestion.
The OP has an infinite money supply and has bribed various officials. As LiveOnAPlane and Exapno Mapcase have mentioned, I’d regard a supersonic fighter with inflight refuelling the fastest. The OP could just go to the USAF and say “Here’s a billion$ , get me 12000 miles as fast as you can.”
But how far can a fighter with external tanks go between refueling. Can a fighter with external tanks go supersonic? Is too much time wasted refueling; would it be better to fly slower allowing more miles between refueling? The fighter would need to be a 2 seater; that limits the available airframes.
After a quick poke around at Wiki at the F14 (retired,WTF!!), F18 and F111, it seems that the first two have a ferry range of about 2000 miles, with the F111 with 3200 miles. But there’s no data on the speed to achieve these ranges.
So all I’ve done is pose a bunch of questions; perhaps if they’re answered we’d be closer to a solution for the OP.
Or find that some else has a more realistic answer 
I´d put my money on a refurbished SR-71, if you drop the spooking gear and kick the second guy out to get more tankage you could squeeze some more fuel, maybe enough for a range of some 8000 Km (nominal was about 5500), three refuellings would be necessary and at a WAGed average speed of 3000 Km/h you could be in the antipodes in about 7 hours.
Fast enough?
According to the Wikipedia, the Space Shuttle can go at least 17,180 mph. So you could not only make the ride in an hour or two, but unlike the ICBM idea, you would be able to land too.
…at the nearest site capable of accomodating the shuttle, which might still leave you some distance away. In any case, that figure is for orbital velocity, not for travel through the athmosphere.
Well as another thread recently showed, the exact opposite side of the world is most likely in the middle of the ocean, so no craft whatsoever could land there. (Some sort of boat would thus be your answer.)
I was assuming that the OP wasn’t feeling particular about exact semi-circumnavigation.
Yep, but I suspect that it doesn’t take all that long to get to that speed. The main question would be whether it could go halfway around the globe without having to leave the atmosphere, or if it did whether it could go up and down in time or if it would have to go around 1.5 times or such to get the proper curve.
Why a fighter jet? Wouldn’t a bomber be faster, all things considered? If they can do the trip without refueling, I bet that would make the difference.
Still, Triskadecamus has it. By the time you are done talking your way into a military jet, your commercial flying competitors are already halfways there, if not there already.
Like in reply #5? 
(Edit: hang on a minute, can’t we just put an aircraft carrier there?)
More use of Wikipedia: A ‘transoceanic abort landing’ will get you from Florida to Africa or Europe, but not beyond. (Perhaps a deliberate part of the shuttle design, that this abort mode occurs while large land masses are available?) “Although a TAL abort has never been necessary, the huge velocity involved means that the total time from liftoff at Kennedy Space Center to landing in Europe would, according to one astronaut, take ‘less than 20 minutes’.” However, the antipode you’re heading for is near Easter Island, so you’ve not really made much headway. What you’d have to do would be land from orbit at Easter Island itself, which is capable of being a shuttle emergency landing site. How long a journey will this be? No idea.