Generally the answers here have been success stories, so I thought I'd share my recent experience sitting just outside such a project that did not go so well.
How did you fund it?
How did you staff it?
The project started when my manager identified a potential employee, lets call him Fred, who had a pet project in our field. We don't pay well, and they they agreed that Fred would be hired and would work almost full time on the project, which they would eventually introduce to the business.
So Fred's started work on the project, known only to Fred's team but not to management or other parts of the business. Fred is a developer, and the work was more-or-less pure development, plus contributions to an underlying open-source project.
Was it successful?
Not really. Fred was working on it alone, and I think would have spent 12-18 months on it. Progress reports to the team consisted of describing whatever bug he was fixing that week. Occasional attempts were made to interest one or two higher-ups in the organization, but they never really went anywhere. Fred was supposed to put together a plan to finish and roll out the project so it could be introduced to the organization, but there always seemed to be some reason it was never done.
Were you found out?
Word slowly filtered out as Fred an the manager tried to interest more people in what they were doing.
Eventually we got restructured, and our new director wanted to know what everyone was working on, and the project was revealed to him. However, it was apparently not explained very well, since the new director wound up asking me (and others in our team I am sure) what exactly Fred's project was?
Were you punished or rewarded?
Eventually the new director froze all funding for the project and Fred was reassigned to work on other projects. That's the current status as far as I know.
How long did it take to finish, compared to above-ground projects?
Was it formally adopted?
It was not finished and it was not adopted.
What was the cost, compared to above-ground projects?
The ostensible cost was Fred's time.
However, there were other costs.
First, Fred and his project became a it of a joke in our team, and later in the teams we work with. What was he doing? Why was he doing it? Why was there no progress? Fred's reputation suffered. "Fred's project" became an in-joke for a project that was going nowhere.
Second, the eventual revelation of such a long-running but hidden project reflected poorly on our manager, and by extension on our whole team.
Third, resentment grew. Why was this guy working on his pet project when there was so much real work to be done? We are a small but busy team and we could have used a developer on any number of other projects.
In the end, I think this project has had consequences for our team's standing and dynamic. I occasionally talk it over with team members, when we're away from the office. Initially (and at the time) we were very critical of Fred, who can be an irritating guy, and who does not take criticism well, and who promised something he couldn't deliver. More recently, we've been critical of our boss. This was not a good way to run a project, and it was obvious from very early on that Fred did not have the skillset to do this work on his own and he would not seek or take advice. It was unfair to Fred that he was put in that position and left in it for so long. Lately I have wondered if I should have raised my concerns more forcefully. Though we did push Fred and our manager on what the project was and where it was going, we did not take it any further than our team. Having said that, I cannot imagine a good outcome even if we had.
Finally, I'd like to say that Fred is a smart guy and the project was not a bad one. It could have been successful (some parts have since come out in competing projects -- inferior competitors that actually delivered).
If this project had been done above board, and Fred had been working with a decent project manager and had a good communicator on the team, it could well have found a champion and delivered something great. Either that or it could have been killed a lot sooner.