Hi, you will hardly reach any practical limit either by having one catalogue with all your clients images or having a separate catalogue for every client.
Having said that, I would also recommend working with one catalogue, unless there is a specific reason to do it (there are a few). My other recommendation is to put the LR catalogue and LR cache in a SSD or high performance disk.
Ligthroom catalogues (*.lrcat files) are self contained databases (SQLite databases) where the data is organized in tables and can handle large amounts of information. There is a theoretical limit of 140 TB, which I don't think should worry any photographer at present times.
As previous posters said, you can separate your clients by using appropriately the collections, keywords, etc.
Why would I use more than one catalogue? First reason is if you work in a multi-user environment where concurrent edits to images occur. Without getting too technical, SQLite is not adequate for multiuser updates. Even in these environments, you must consider first the option to export the images to be edited by another user as catalogue (from the main catalogue) and then import the updates when finished.
On the other hand, if you really want to stick with your idea of having individual catalogues for every client, then the limit is what your OS can handle (since every catalogue is a single file), practically tens of millions in every folder, so again nothing to worry about.
If you go this way, you'll lose the ability to group and search through all your images at once and increase the risk of mismanage the files, not to mention the backup and LR update version processes which can turn into nightmares.