For processing of Landsat, I can recommend GRASS. I tried many others.

You may need to refine your question with regard to the type of imagery you propose to use. There are workflows which have been more or less developed and implemented in various software.

Not only the type of imagery, but the purpose of the processing and final analysis. For Landsat, I am interested in a quantitative value. Which is different to qualititive methods used in regional classification of vegetation for instance, methods and tools for this work are more common.

You will not likely find a Swiss Army Knife for free. But you will find very specialised tools which do one job well.