The best replacement for anniversary is birthday.

The best alternatives to anniversary are rated as following:
★★★★★  birthday
★★★★★  underworld
★★★★★  jubilee
★★★★  annual
★★★★  anniversary
★★★  halo reach
★★★  reach
★★  monthsary
★★  tomb raider 1
★  happy birthday
★  wedding anniversary
  10th year anniversary
  halo combat evolved
  first anniversary
  halo 3
  wedding
  1926
  world class
  tomb raider legend
  golden birthday

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Speaking about anniversary, one probably shouldn't forget about panasos.

Other search terms related to anniversary and their ratings are:
★★★★★ panasos
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anniversary
birthday
underworld