According to code, the Client Response Gallery is capable of exporting up to 50,000 images! Of course, it would be absolutely insane and completely irresponsible to create galleries of such colossal weight. Computers and handheld devices have limited memory, and web-browsers can only handle so much. There is a breaking point, and beyond it your galleries will be unusable. And so it's important to be wary of functional limitations, careful not to load too many images into a single gallery.
What's the limit then?
I would never publish a gallery in excess of 1,000 images, nor even approaching that number. But some photographers have done so and gotten away with it. Still, I don't recommend it.
Keeping usability a high priority, allowing a safe margin for error, and accounting for clients maybe having a shorter attention span for image evaluation than you do – images are your passion, and may not be theirs – I consider 500 images to be a fair “soft” limit for an image gallery. If my collection is larger than 500 images, I will typically break it into several smaller collections, published as separate galleries, then use TTG CE3 Auto Index to corral those galleries under a single index. This prevents undo stress on the hardware, and keeps galleries to a digestible size for clients.
Most jobs will lend themselves to natural break points. For example, rather than create a single gallery in excess of 1,000 images for a wedding, the day might be broken into separate galleries for pre-ceremony, ceremony, formal portraits, and reception. A fashion shoot might be divided into separate galleries for each outfit, location or model. Etc.
Even if testing a largish gallery on your computer and you feel it runs adequately, bear in mind that your client may be viewing on a mobile phone or tablet, which have far less memory than a desktop computer. Because they have limited memory, mobile browsers are more prone to crashing under the weight of heavy web pages, which could lead to a very frustrating or impossible situation for your client attempting to make image selections. They may also be loading your galleries over a cell network, rather than WiFi, exacerbating the problem.
The Client Response Gallery has existed in various incarnations for many years now, and very few photographers have taken significant issue with the gallery's limitations. In fact, many photographers – some of them well-known, such as Terry White, Lee Varis and Scott Kelby – have found it to be a valuable addition to their photographic workflow. It's just important to be wary of the gallery's capabilities and limitations.
If may be helpful to publish test galleries, to test those galleries on various devices, and to get a feel for how large a gallery you are comfortable publishing for your clients to use. You may find that you can get more mileage from the gallery by publishing images at smaller dimensions or at a slightly lesser quality value, or you may find greater success splitting collections into smaller groups and publishing multiple galleries, as I do.