It doesn't stop. Next step is a mouthpiece with a wider tip. And then more reeds...
Yes, that's about the size of it. The original poster will find that things do settle down after a bit. While there are exceptions to the rule, most players tend to standardise on one mouthpiece and one reed combination. Other players will have a couple of mouthpieces e.g. one general purpose mouthpiece and another which is more punchy for live gigs e.g. perhaps a Dukoff. Take me on tenor - I have an 1970s "Early Babbitt" Otto Link STM for standard use (it's lovely!), an Otto Link Tone-Edge if I need something darker, plus a Rico Metalite for those rare open air gigs, where I'm being drowned out by the rest of the band and need to burn the paint off a nearby fence. However, upwards of 90% of the time I'm playing that Otto Link STM.
A few players will have 3 or 4 mouthpieces in their collection. However, it's all too easy to become a jack of all trades and master of none. That's why ueber-professionals such as Michael Brecker (who started on an Otto Link STM in the 1970s and then moved onto a custom-made Guardala) tended to stick to one particular mouthpiece so they could achieve total mastery of their instrument.
That's not to say that established players won't experiment and make occasional forays at a major sax-shop to try something new for 20 mins. The quest for the perfect mouthpiece is something which tempts every player just a little bit. Who could resist the temptation that somewhere on this planet is a mouthpiece which will somehow make your sound "click" neatly into place and give you everything you ever wanted and more besides? Trouble is, that search can quickly become like the quest for the holy grail if you're not careful. You can end up going round and round in circles, then finding out that the mouthpiece you already had was actually pretty good.
Consider:- there are probably around 12 professional-grade saxophone manufacturers which have ever existed e.g. Yamaha, Yanagisawa, Keilwerth, Mauriat, Buescher, Conn, Martin, King, Couesnon, Holton, Dolnet, Beaugnier, Selmer and Kohlert. If you factor in the different models which each manufacturer made (such as the Yamaha YTS-62 or YTS-875 etc) then you probably end up with around 60 horns to choose from. Obviously, we're including older horns such as a late-1920s Conn New Wonder Series II in that mix - but it's still a good selection to choose from.
However, that figure of 60 is dwarfed by the many hundreds of different mouthpieces out there. For example, let's assume you went for an Otto Link. Well, they do at least 3 different models of mouthpiece - the flagship being the Otto Link STM. Even if you pick one of those, you still need to decide whether it's the standard STM or the STM NY. Having made that choice, you now have to select the tip opening - and there are plenty to try - from 5 through 9* in half-increment steps. So, that's 10 different options. In other words, for Otto Links alone that's well over 40 different mouthpieces if you include the STM NY.
Finally, do you accept the factory-made Otto Link mouthpiece "as is" or get a mouthpiece guru to modify it for you. Alternatively, do you pick a vintage model e.g. a 1950's vintage Otto Link, or a 1960s "Floridian" Otto Link or an 1970s like the one I play? From 40 different mouthpieces made by Otto Link the figure can suddenly increase to around 80 options for Otto Link mouthpieces alone. Oh yes, it quickly gets complicated.
Now, when you bear in mind that Otto Link is just one of many mouthpiece manufacturers out there (including Jody Jazz, Meyer, Dukoff, Vandoren, Theo Wanne, Aizen, Guardala, Ed Pillinger, Beechler, Selmer, Bari, Brilhart, Lakey, Berg Larsen, Ponzol, Lawton, Rousseau, Yamaha, Yanagisawa, Runyon, or various Chinese clones etc) the permutations quickly multiply into the thousands. Jody Jazz alone has manufactured a total of 7 different mouthpieces over the years, with 4 still in productions. New mouthpiece manufacturers spring up every 5 years or so. All you need is a CNC machine, lathe and bench-press drill, an assortment of files and a dream. Any competent machinist can make a saxophone mouthpiece that works and is OK. Of course, the trick is knowing how to make really good ones that sing.
The big risk is that you spend so much time searching for the holy grail of the perfect mouthpiece, that you don't have time to actually play the damn horn. And that is why, apart from the occasional experiment from time to time, most players standardise on one or two mouthpieces. I have 3 mouthpieces for tenor. Two of them spend most of their time gathering dust. The other (1970s Otto Link STM) is the one that gets played.
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