If I am not mistaken this is not possible at the moment at project-gc. In principle it is of course possible to generate this anniversary list (let's say all caches in a given country/region/county that have their anniversary in the next week or so). But in order to cache this data (to keep the site performing fast), one would need to cover a lot of different variables (which anniversary, how long in advance, which region, etc.), so it seems to be at least a bit tricky.
Although it is a bit of a work, it is possible to do this with the Pocket Queries at geocaching.com. Here you can ask to show you caches placed in a given time range. So you can select the anniversary range (but you have to do this for every year above the ten year limit, and you have to change the date range every time you will be looking for it; hence the laborious amount of work). The upside is, though, that caches that old are not abundant in general, so you can use a large area for the pocket query (easily a country or even multiple countries).
Hope this helps.