Are any agile (aerobatic) private jets produced? If you look at the car industry there are luxury comfort brand like Rolls Royce or Bentley but super expensive racing cars are probably even more popular. They offer limited comfort, are not practical, almost not usable on normal roads, not very reliable and offer terrible fuel consumption. But still everybody dreams about them and a lot of companies try to manufacture.

Business jet can offer luxury and comfort but what to do if you want some excitement or performance? Why are there no jets with massive after burning engines with thrust bigger than weight that can climb straight up and do several G? Something like a cross-breed between a Gulfstream and an F15? Maybe it's good business case to manufacture something like this?
The reason these aren't built is that there's no market for them. Sure, there's loads of people who would love one of these (including me) but they would be incredibly expensive to buy and operate, far out of reach for anyone but the extremely wealthy. 

At the airfield I fly from there are Jet Provosts, a 50's era jet trainer, available. These old, not particularly fast or agile jets cost about 1000 USD per  hour to fly. I've seen Mig-21 (certainly fast with lots of Gs) training for about $4500 per hour, which gives you an idea of the running costs for a fast, exciting jet. 

The existing market of people who can afford this is already supplied by military surplus jets. These run from trainers like the L-39 and Gnats to fighters and bombers like Migs, an F-104, hawker hunter, there's even a privately owned Harrier. These don't cost that much to buy considering what one would cost new, and there's plenty of them on the market, so there's no call for new builds.