Welcome SanFranDan!
There's a well documented alternative known as the Yamanote Challenge where those taking part walk the entire Yamanote line circuit gathering evidence from each of the stations to show they visited.
A fairly fit person will take around 12 hours to walk between all 29 stations.

Some of the stations will have an ink stamp somewhere so if you take a pad of fresh rubber stamp ink to use (the ones at the stations often run dry) you can collect a series of seals. Commemorative stamps appear all the time, they are often called "stamp rally" should you need to ask for them - helps if you have an example already to show when enquiring.
Not on the circuit but Ryogoku station has yielded my favourite stamps so far.

Just checked by plugging yamanote line stamp rally into google and it shows there was a Pokemon theme going on in 2011
http://www.webmtabi.jp/201108/pokemo...e-line_en.html
More info here:
http://my.opera.com/Nplus/blog/show.dml/400692
I don't think most of the participants walked...