Hi - Great question! The good news is no! Best practices say keep a handful and not more. Think of it this way... When you need a backup to restore from you will generally want the most current backup because it will reflect your most recent work. Also you don't have to backup every time - especially if you haven't made a lot of changes in your catalog file. And trusting that you know that Lightroom isn't backing up your photos... only your catalog.
It may be that you have your LR set to backup every time it closes - and if you allow it to do this, you will soon accumulate a large number of old copies of your Catalog. I agree with another poster that the most useful backup to keep around will normally be the very latest, since that can retrieve the maximum work should something catastrophic happen.
Sometimes a Catalog gets subtly corrupted or else simply some unintended edits occur, and this is not discovered straight away. So it's a good 'backstop' to retain at least one non-recent backup as well. If your recent backups turn out to all share these same problems, you then have got another option to call on, which probably will not. And you can at least seek to reinstate more recent editing, from a sound foundation.
If you have had "Automatically Save Changes to XMP" turned on while you worked, there is then the possibility of retrieving some of that work even if carried out subsequent to whichever backup you wind up using. For images which were already imported when that backup was made, you can load in subsequent editing of those using Read Metadata from File. For images which were not imported at that time, those files when imported again will show their latest prior editing. This works only for master copies not Virtual Copies, and only reflects those sorts of metadata which can be written out to XMP on the file (so Develop adjustments, Snapshots, keywording etc yes; Collection membership and such no). This is not a substitute for making Catalog backups, but sometimes a useful supplement.
It is worth pointing out too that a backup which is saved onto the same storage drive as the working Catalog is, can therefore be of no protection if this whole storage drive becomes unreadable / fire damaged / stolen / whatever. Conventional file backup of all your images, regularly refreshed / cycled, into a physically separated system is always a good idea and this can advantageously include your working presets etc, your Catalog and a selection of Catalog backups. It does not need to include LR cached previews / smart previews since those are easily recreated.