ADPro, DPaint, Ced, AmiTCP. AmigaVision, the Gold Disk creativity apps, Lightwave and the Toaster tools, obviously. LHA/LZX, as well as other similar type shareware utilities. Too many games to list. SAS/C and the whole plethora of different programming apps and utils of virtually every programming language that made so much cool software possible.
MUI and the apps it ushered in were a biggie too. In the infancy of the consumer availability of the Internet, some of the MUI apps from VaporWare were huge. Miami was a biggie for the common Amiga grunt back then when dialup Internet was just becoming popular, made connecting real easy compared to fighting with AmiTCP 
Let's not forget what came prepackaged - ARexx! Allowed the Amiga to do some pretty darned amazing things for the time.
Again, what further innovations did these apps influence? Also, neither LHA/LZX or Rexx were Amiga "firsts." How about I say that Doom is one of the most influental Amiga apps?