The best replacement for jpeg is tiff.

The best alternatives to jpeg are rated as following:
★★★★★  tiff
★★★★  raw
★★★★  jpg
★★★  png
★★★  gif
★★★  pdf
★★  jpeg2000
★★  bmp
★  bitmap
  tif
  mpeg
  png smaller
  gif for web
  raw images
  nef
  raw format
  dng
  pdf which is better
  bmp quality
  jpeg 2000
  jpg difference
  raw files
  tiff files
  pdf for printing
  webp
  bitmaps
  pdf files
  bitmap images
  tiff which is better
  pdf file size
  pdf quality
  bmp files
  gif images

Your alternative for jpeg is not listed? Suggest!
Speaking about jpeg, one probably shouldn't forget about transparency.

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



Google Trends

Scale is based on the average worldwide traffic of jpeg in all years. Click on graph for more.
jpeg
tiff
raw