Part of the difficulty with tabooing "self" is that tabooing it would require replacing it with a clear definition, but I'm not sure that the concept is coherent enough to have a clear definition. Also, different people mean different things by it.

Anyway, the point isn't that you wouldn't exist as a physical entity. You certainly do. Some possible meanings of the no-self claim are:

No-self as the lack of a homunculus: there's no self in the sense of a unitary "master controller" in the brain that would be active all the time. Rather there are just a variety of modules, with some set of them being active at one time, another set of them at another. E.g. Kurzban & Aktipis in Modularity and the Social Mind: Are Psychologists Too Self-Ish?:

What if there were no unitary self to be “interested,” “deceived,” “regarded,” “evaluated,” “enhanced,” “verified,” “protected,” “affirmed,” “controlled,” or even “esteemed”? If there were no such self, what should we do with theories such as those that make reference to self-affirmation (Steele, 1988), self-evaluation (Tesser, 1988), or self-verification (Swann, 1983, 1985)? If there is no singular “self” that is meaningful in the context of theories that use this term, it might be time to rethink the areas of inquiry these theories address (Kurzban & Aktipis, 2006; Rorty, 1996; Tesser, 2001; see also Katzko, 2003, for a recent discussion).

Here we propose that the ontology of the self is deeply connected to the issue of the extent to which the mind is modular, consisting of a large number of functionally specialized information-processing devices, each of which processes only a narrow, delimited set of inputs (Barrett & Kurzban, 2006; Coltheart, 1999; Cosmides & Tooby, 1994; Sperber, 1994). Modular architectures result in systems that are potentially computationally isolated from one another. This makes statements about “the” self problematic—what, precisely, is the referent (Leary, 2004a)?

See also TEDxYouth@Manchester 2011 - Julian Baggini - Is There A Real You? for a related view.

No-self as the lack of continuity: the teletransportation thought experiment is possibly the best illustration. Suppose that your brain and body are scanned into a computer with perfect resolution and detail (presume, for the sake of argument, that this would be possible). Your original brain and body are then destroyed, with a duplicate being manufactured somewhere else. Were you transported to some other location, or was the original you murdered with a copy manufactured in its place?

The "no-self" view says that if all of your data was perfectly copied and the replica is indeed perfect, then there's no meaningful way by which you can claim to be different than the "original", and have no reason not to consent to the operation. The "self view" would say that you should never consent to such an operation, because you get murdered in it, and although an identical copy of you does get constructed, it isn't you but merely a perfect copy of you. The "no-self" view mutters under its breath that the "self view" seems to be postulating some kind of an intangible soul or something.

The "no-self" view might also claim that even now, the you who is reading these words is a different person than the one who hadn't yet started reading this comment. Very similar, yes, but your brain now isn't in the exact same state as it was before: it has changed, and therefore become another person. Even if the teletransporter did kill you, that doesn't matter, since you get killed and reborn all the time anyway.

The phenomenological no-self. We generally think that experiences are experienced by some specific person. If you could choose between your loved one or a total stranger experiencing some amount of suffering, and your choice would have no other consequences, then all else being even, you would probably prefer that it was somebody else than your loved one who had to suffer. Though maybe if you knew that the stranger had already suffered a lot in their life, you might decide to spare them the suffering, because they'd had enough of it already.

The "no-self as the lack of continuity" view might say that this choice is meaningless, because neither your lover nor the stranger are continuous people - even if the stranger had already suffered a lot, that suffering had been experienced by a huge number of other people, not the one who will suffer the consequences of this decision. But even if your lover wasn't a single person, but a collection of many closely related ones, you might still choose to care about that collection of closely related people not suffering, in which case the "no-self as a lack of continuity" view wouldn't be relevant for your decision.

The phenomenological no-self view, however, takes things one step further, and asks what it even means for something to be experienced "by" someone. Yes, an experience is clearly the result of some kind of information-processing happening within an organism, and that information-processing is (in a functional and a physical sense) happening within some particular organism. But what about the phenomenological experience, the qualia of sensory experience - what does it mean that that's experienced by someone? It would seem to imply that there is a specific observer who observes, for example, sensations of pain. But what, exactly, is that observer? The simplest answer is that there isn't one - sensory experiences aren't experienced by anyone in particular, they just happen. Ken Wilber, in No Boundary:

Begin with the sense of hearing. Close your eyes and attend to the actual process of hearing. Notice all the odd sounds floating around—birds singing, cars rumbling, crickets chirping, kids laughing, TV blaring. But with all those sounds, notice that there is one thing which you cannot hear, no matter how carefully you attend to every sound. You cannot hear the hearer. That is, in addition to those sounds, you cannot hear a hearer of those sounds.

You cannot hear a hearer because there isn’t one. What you have been taught to call a "hearer" is actually just the experience of hearing itself, and you don’t hear hearing. In reality, there is just a stream of sounds, and that stream is not split into a subject and an object. There is no boundary here.

If you let the sensation of being a "hearer" inside the skull dissolve into hearing itself, you might find your "self" merging with the entire world of "outside sounds." As one Zen Master exclaimed upon his enlightenment, "When I heard the temple hell ring, suddenly there was no bell and no I, just the ringing." It was through such an experiment that Avalokiteshvara is said to have gained his enlightenment, for in giving awareness to the process of hearing, he realized that there was no separate self, no hearer, apart from the stream of hearing itself. When you try to hear the subjective hearer, all you find are objective sounds. And that means that you do not hear sounds, you are those sounds. The hearer is every sound which is heard. It is not a separate entity which stands back and hears hearing.

The same is true of the process of seeing. [...] The more I try to see the seer, the more its absence begins to puzzle me. For years it seemed perfectly natural to assume that I was the seer which saw sights. But the moment I go in search of the seer, I find no trace of it. In fact, if I persist in trying to see the seer, all I find are things which are seen. This simply means that I, the "seer," do not see sights—rather I, the "seer," am identical to all those sights now present. The so-called seer is nothing other than everything which is seen. If I look at a tree, there is not one experience called "tree" and another experience called "seeing the tree." There is just the single experience of seeing-thetree. I do not see this seeing any more than I smell smelling or taste tasting.

It seems that whenever we look for a self apart from experience, it vanishes into experience. [...] As you are now thinking about this, can you also find a thinker who is thinking about this?

That is, is there a thinker who thinks the thought, "I am confused," or is there just the thought, "I am confused"? Surely there is just the present thought, because if there were also a thinker of the thought, would you then think about the thinker who is thinking the thought? It seems obvious that what we mistakenly believe to be a thinker is really nothing other than the stream of present thoughts.

Thus, when the present thought was "I am confused," you were not at the same time aware of a thinker who was thinking, "I am confused." There was just the present thought alone—"I am confused." When you then looked for the thinker of that thought, all you found was another present thought, namely "I am thinking I am confused." Never did you find a thinker apart from the present thought, which is only to say that the two are identical. (More.)

(I subscribe to all of these no-self views.)

(Posting this as a separate comment, since adding it to my other one would make it too long)

Note that in each case, the "no-self" view is really more of an antiprediction than a prediction - it only says that some postulated "self" (either in the form of a central controller, some mysterious immaterial essence that you have but a copy of you doesn't have, or some observer entity that is separate from what is observed) doesn't exist, because we have no reason to presume that it exists. People tend to assume a self by default, and ask for evidence for why one wouldn't exist, but to do so is to privilege the hypothesis. A supporter of a "self view" should provide some coherent definition of a self first, and then some evidence for postulating one in the first place.