Whereas architectural projects and deadlines come and go, being able to add software licensing as needed seems like a really good thing. Autodesk has started to offer methods to rent Revit within their Building Design Suites.
Standard Building Design Suite
$360 Suggested Retail Price (SRP)
Premium Building Design Suite
Ultimate Building Design Suite
Monthly plans can be purchased only through the online store and quarterly/annual plans can be purchased only through a reseller.
I haven't found any mention of just the individual flavors of Revit (Architectural, Structural, MEP) being able to be rented, yet. There is AutoCAD Revit LT Suite at $95/month, but its Revit files are not compatible with the full version of Revit as far as I know. (Please correct me if I am wrong.)
Here is the chart of what software is included in the Building Design Suites: http://www.autodesk.com/suites/building-design-suite/included-software . Mainly, the Standard one does not have Max or Navisworks with it.
This is the Autodesk page with the rental prices: http://www.autodesk.com/suites/building-design-suite/buy . They use the term "PAY AS YOU GO" and don't write "RENT" on that page.
UPDATE: The latest page for renting and buying Autodesk software is at http://www.autodesk.com/store/suites . At Autodesk University 2014, they announced their plans to enable a "Subscribe to Autodesk" way of paying for everything at once.