The best replacement for arcgis server is arcims.

The best alternatives to arcgis server are rated as following:
★★★★★  arcims
★★★★  geoserver
★★★  arcgis online
★★  arcsde
★★  mapserver
★  spatial data server
★  arcgis
  mapguide

Your alternative for arcgis server is not listed? Suggest!
Speaking about arcgis server, one probably shouldn't forget about openlayers.

Other search terms related to arcgis server and their ratings are:
★★★★★ openlayers
★★ arcsde
You have in mind a companion for arcgis server and it is not yet here? Propose!



Google Trends

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arcgis server
arcims
geoserver