In my experience, it takes around a year to sell software to a large company. I have seen it getting done in three months, but it is very seldom. Therefore, my question is whether you know of any software, which you believe could get sold to a whole department within one week. My reason for asking is that I am interested in creating or acquiring such software and use it to gain a first presence in the company. From there, I could sell other software so much easier by calling them upgrades, plug-ins etc.
EDIT: I am talking about software, which runs locally in the enterprise, and not a cloud solution.
Probably something that an individual would buy and that is better if their colleagues use it (network effects). I'm thinking of things like Xobni or Evernote but I can't come up with too many examples because I use very little desktop software. Can you break one one small feature of the product and sell it on its own?
Does it really need to happen in one week though? Instead of one week of active selling, what if you provided a stream of information (such as a blog) that would deliver value and eventually (over 3-9 months) convert visitors into buying an introductory product or the main product you want to sell?
The only time I can imagine a scenario in which an Enterprise would purchase software within a a one week period would be a solution to a critical bleeding pain: A security breach, a pending SEC violations, a BI solution which can confirm/deny internal nefarious activities, operational software for managing during a flood/whiteout/radiation leak.
You get the idea.
Enterprises companies purchasing takes a long time because -- well, there are lots of reasons and that wasn't really the question -- but let's just say there are lots of them. Some are entirely reasonable and understood. Others, well not so much. No matter what the reasons, they are there and very difficult to "end run."
Absent a "bleeding pain" solution -- I believe the up-sell from a personal solution to an enterprise level solution as mentioned by @richardg and @elie is your best bet in this innovative and creative marketing/sales strategy.
Creating an individual version of the critical function of your application and make it usable within a corporate firewall -- ideally so the user will not even need to secure permission from the IT gatekeepers. Become essential-- and let them be your advocates for the sale of an enterprise version.
You are always going to run into spending authority limitations. If you want to sell quickly, you have to sell to someone who has the budget and can make the decision alone. You also have to be able to get to this person. So my quick answer is get your price in line with this reality and you will have the best shot at selling quickly.
At one company where I worked, a bunch of the developers went and bought copies of Beyond Compare for doing source code comparisons. Eventually, the company bought a commercial license, but until the process ran its route, single people in the company were already paying for it out of pocket because it worked.
Not a fast approach, but as richardg said, it can get you into a company, perhaps already making money, and pave the route for the full version of the software.