I do not know of any free software continuous backup solution. An interesting design was written up by liw which seems to correspond to what you are looking for, but that design has yet to be implemented.
Nevertheless, here are what are the interesting free software backup alternatives right now.
Dejà dup is a frontend for duplicity so it supports incremental and encrypted backups. It can also keep previous backups. It has some remote servers options and integrates well with the desktop.
Obnam is quite interesting and seems to fulfill all requirements but the desktop support.
Bup and Attic are also interesting alternatives, both of which are really high performance (so that you can run backups frequently, although not continuously) but only
Bup have some (third-party) GUI. Note that Attic is an abandoned project, with a more active fork called Borg and Bup is self described as 'very early version'.
Timevault is one alternative, but it has been unmaintained for a while, and backups only on local filesystems.
Back in time is an alternative that seems to be more maintained but has similar limitations.
Flyback is yet another alternative.
git-annex, while not a complete backup solution, is a nice solution for large file collections, and seemlessly integrates with git, rsync and other tools. It can easily track multiple copies and make sure that you have more than N (configurable) copies of your data. The downside is that the UI is limited and it will fail backing up certain files, like .git directories, ironically enough.
camlistore is similar to git-annex, but is mostly aimed at developers at the time of writing.