I am pleased to note that the must-have accessory for the eccentric dot-com zillionare mogul is no longer the 100-foot yacht, but a private space program!
Currently in the lead would be
SpaceX, founded by Elon Musk, who made his gazillions starting PayPal and selling it to Ebay. They are building a family of boring but cheap traditionalish boosters, and have already reached orbit with their smallest model. (Musk's other current projects are the Tesla Motors electric car company and Solar City home solar array company; he would appear to have escaped from a Ayn Rand novel.)
Cloaked in lane two, we have Jeff Bezos' (of Amazon.com)
Blue Origin. We know that they tested a large VTVL craft a year or two ago, and that they have an engineering facility in Seattle and a manufacturing/launch test site in Texas. Also, they have a coat of arms, with turtles. Outside that... (shrug).
And last we have the darkhorse crowd favorite, John Carmack's
Armadillo Aerospace. Their budget is at least an order of magnitude lower than the others (which is to say, in the low 7 figures), and most of their work to-date has been volunteer labor while John continues on his day job as lead coder for ID sortware (is that Doom 2 or Quake 4 now, I forget?). His frequent blogging is tremendously entertaining: Feburary's 'Lessons learned' included "Use the right bolts." and "Set up our big 1600 gallon fire tank with two fire hoses so we can cover both sides of a fire simultaneously." It's easy to dismiss them because of their small scale, but they've made something of a virtue of it, ruthlessly maintaining simplicity while gradually adding capabilities and reliability - they now have single-stage rockets that can take off, hover for 2 minutes and land quite precisely. They even have a mad but not obviously infeasible plan to scale up by building a massive array of cheap modular rockets (RAIR?)...