on average, using only transcendence and no crown of experience, 1 in 276 items starting from +0 will make it all the way to +8. However, you'd want to be using Crown of Experience from at least +6, and if you use it all the way, 1 in 55 items will make it all the way to +8. That said, it is not the intention that you will succeed at upgrading your items that far, generally, and also many weapons start around +4, where around 1 in 28 items will make it to +8. The primary purpose of the system is to serve as an item sink/filter, and originally items could only be upgraded to +5, with transcendence enabling them to get upgraded to +6, but that proved to be too common to serve as a proper item filter. In an ordinary playthrough, when you've familiarized yourself with the system, chances are you'll see a large amount of items hit +5, a good amount hit +6, and a small handful hitting +7, with +8 being rare occurrences though it does seem to occur sometimes into hardmode. This will probably alarm you to hear, but I am actually planning to extend the upgrade level system further in the future.
The purpose of making crowns rare is not to serve as a loot grinder, but to force you to reserve them for items that are difficult to acquire and which you're going to actively use. Even among those, you are meant to hold back a bit until the endgame, ie. probably not upgrading Night's Edge and Excalibur beyond +6 or so, but weapons like Terra Blade you might take the whole way. In a sense though, the primary purpose of the system is to counter the loot grinder, that is, give you something to do with all your loot by throwing it against itself until the pile is much smaller (up to 276 times smaller in this case)