“Stickyness” is a term of art usually applied to websites like Facebook that keep visitors coming back for more.

But it can also be applied to a software platform, and in a note to clients issued Friday Deutsche Bank’s Chris Whitmore takes a crack at measuring the stickiness of the iPhone-iPod-iPad platform.

What he’s done is analyze the size of the iPhone, iPod and iPad installed base and the investment Apple’s (AAPL) customers have made in content for those devices in terms of apps, videos and music purchased at the iTunes Store.

The result is a chart that shows two things: First, an Apple installed base today of about 150 million iTunes-dependent devices (iPhones, iPods, iPads) that could grow to more than 200 million by the end of 2011. And second — using Apple’s “Other music related products and services” revenue line to estimate sales of music, apps and videos — a cumulative investment in those devices that stands at about $15 billion today and which Whitmore sees growing to $25 billion by the end of next year.

“This averages to ~$100 of content for each installed device,” he writes, “suggesting switching costs are relatively high (not to mention the time required to port). When Apple’s best in class user experience is combined with these growing switching costs, the resulting customer loyalty is unparalleled.”

In light of what he calls “the stunning success of the iPad” and the demand for upgrades that the new iPhone is likely to spur, it’s going to get harder, not easier, for users to switch, no matter what Google (GOOG) and the rest of Apple’s competitors have up their sleeve.

Below: Two Deutsche Bank charts. 1) Average user investment in content per device, March 2007 to December 2011, and 2) Estimated iPhone, iPad and iPod touch shipments, June 2007 to September 2012.

[Follow Philip Elmer-DeWitt on Twitter @philiped]