There are numerous opportunities in language design, implementation, and run-time systems to take advantage of IEEE 754 features.
Uninitialized Arrays
Initializing arrays to NaNs (perhaps all 1's instead of 0's or old contents [security risk!]) actually made it into C9X discussion.
NaN encodings of PC, hashtable entries etc
Clever trap handling and retrospective debugging tools (interactive or post-execution?)
Missing Data
(Make sure it is not used!)
Most languages do not provide tools for such encodings, e.g. allow no access to NaN field, or even refuse to have more than one configuration of NaN.