Besides GIS applications, which other applications or libraries use R-trees and its variants?
Computer games often do. Here's a link to something cool.
Computer graphics - both software and hardware - often use spatial partitioning e.g. BSP; there's neat logic in Imagination and Intel chips termed tile or zone rendering, for example.
Code I've seen tends to lean towards fixed resolution dividing - octtrees, typically - rather than variable size buckets, but I'd still consider them 'variants' of R-trees.
"R-trees have grown everywhere; from CAD and geographical databases to multimedia and time series management systems." http://www.rtreeportal.org/
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Some relational databases use R-trees for indexing.
PostgreSQL is one example. http://www.postgresql.org/docs/8.1/static/indexes-types.html
Here's an application -- a personal carbon calculator that resides within your smartphone -- using R-trees that are part of the Perst Lite open source embedded database.