Virgin Galactic received approval to carry passengers from the FAA recently, although they are probably some way away from selling tickets.
For a long time the highest-altitude relatively mainstream option was the Concorde, which cruised at around 56,000 ft/17.1 km on transatlantic flights; the vast majority of conventional aircraft fly transatlantic between about 29,000 and 41,000 ft for comparison. This is rather considerably short of outer space, conventionally defined as 1000 km (3,280,840 ft), but high enough for the sky to darken and to see the curvature of the earth.
The Concorde was forced into retirement in 2003, but various large business jets like the Citation X+ and LearJet 75 are advertised with a maximum altitude of 51,000 ft, and the view from 50,000 ft is still pretty cool. They don’t normally operate at this altitude of course, so it would take some legwork to charter, but it’s something definitely in the realm of possibility for a resourceful person in an advanced economy.