How many times can you present the same material at different conferences before it becomes stale?

It's important also to look at your audience. If you're likely to be giving the same talk to two audiences that overlap significantly, then the time between talks is less important than the fact that the audiences will perceive the talk as 'stale'.

For example, you might give a talk at a general interest conference, and then a slightly modified version of it at a workshop with specialists, many of whom may not have attended the general interest conference. That would be fine. You might then go on the colloquium circuit and give the talk at venues with only small overlap with the specialist group.

But if you gave the same (or similar) talk to the same (or similar) audience more than once, that would be a problem.

How long can one talk about the same material in the conference loop until the talk becomes stale?

As other answers have pointed out, certainly the answer is not going to be "N months" for any value of N. Here are some things to think about:

1) You say that "my complete results all reduce to a standard storyline". Yes, but that's true for entire subfields of mathematics. I like to tell the story about how I heard a close friend in grad school give a talk about his thesis work just as he graduated. I heard him speak again when he came back to visit almost a year later. As we were catching up afterwards, I somehow let on my surprise that he gave essentially the same talk both times. He looked at me in disbelief and told me "There was hardly a single result in common between those two talks". It turned out that -- not withstanding that we were both arithmetic geometers -- I simply didn't have enough knowledge in his subfield of speciality to penetrate far enough beyond the standard storyline -- Galois representations, big rings, some filtered semilinear algebra, perhaps an R = T theorem... -- to see what was new about the second talk, despite the fact that an expert would say that it was all new. I have found this to be one of the more profound experiences of my professional life and I tell it often in a variety of contexts: e.g. I mentioned it to a linear algebra student who said that she was largely following the lectures but sometimes was having trouble understanding the differences between what we were doing from week to week. When you tell an undergraduate a parable, it's probably safest to spill the beans and tell them the moral: I recognized her perception as an indication that she was farther away from engaging with the material than she herself realized. In your case the moral is admittedly a little less clear, but I think that part of it is that we can build a highly successful academic career by hewing largely to a "standard storyline": the devil, and thus also the glory, is in the details.

2) So I have a question: is your concern about giving cognate talks or literally giving essentially the same talk over and over again? Let me assume the latter. It's harder to know how often you should simply repeat yourself, and I am wary of giving advice because I know myself to be close to one end of the spectrum: I don't usually travel around and gave the same talk multiple times. The one time I really remember the feeling of repeating a talk more than once was when I was on the tenure track job market, when I did it four times in a couple of months. The last time I gave that talk, it indeed did feel stale and I remember not getting much in the way of feedback from the audience...and then I got offered the job. I am much more likely to want to take every given talk as an opportunity to speak about what's on my mind that month or that week: this is not necessarily a good thing, but I try to make it work.

I think though that you are rightly concerned about giving the same talk to largely the same audience of specialists. When that starts happening, might you not simply be speaking too much? One thing to do is look around: if you're in the same group of specialists, they are also speaking frequently, I guess. Do you observe others around you -- at a similar career stage and also those who are more senior -- largely "repeating themselves" in the talks they give? (Sometimes a really good talk is worth repeating. I remember fondly having heard Manjul Bhargava talk about the "15 Theorem" at least twice, probably three times. Each time I grasped the picture more fully and thus liked it even more. Obviously Manjul could have talked about other work; he chose to revisit this amazingly beautiful well, and his insistence on promoting his own good taste has had such a positive role in influencing later mathematical work, including some of mine.)

Here's a piece of advice: when someone invites you to give a talk, before you accept or decline, ask them what they have in mind for you to speak of (be ready to follow up immediately with a list of options). If you think that the person who is inviting you to speak has heard you speak about the same topic before, or if a lot of the audience feels the same way, ask them explicitly about that.

Another thing to do is to consciously update your "standard talk" to reflect recent changes in your thinking about your research. You're right that a talk is a story, and the story can change even if the core results remain the same. Reinterpreting your past successes in terms of what you're doing presently and are trying to do in the very near future is a big part of research. I have heard James Maynard speak about bounded gaps between primes; I would gladly hear him speak again, even if I were a seamless expert on his paper on the topic (note the subjunctive!) just to hear at the end of his talk what he is thinking about now.

Is a repertoire of multiple research talks expected of a recent PhD?

Multiple disjoint talks? No, probably not. It is not too soon to start working on a "non-research talk", i.e., a much friendlier broader talk that you could give to undergraduates, but I think that's not what you're speaking about. I do think that having multiple projects is a good thing even for a young mathematician nowadays, but one does not want to sacrifice breadth at the expense of depth. If you have basically one research talk that you've given half a dozen times or more over a period of a couple of years, by now that one talk is probably really good. I would concentrate on further refining and updating that one great talk rather than just feeling you should start from scratch in order to show "multiplicity". (Again, I am conscious of being unusually lured by the siren song of multiplicity. Oh, well: we all need to identify the good career advice which is not good for us.)

A recently minted PhD would not typically be expected to have multiple seminar-length talks at her disposal, because in general the PhD work should be presentable as an integrated whole. However, once one starts getting substantially into the postdoc phase and beyond, one should have more than one research talk available, as there is greater diversity in one's portfolio of work.

With respect to "staleness," this is also somewhat of a concern, but only partially. Research progresses in time, but that doesn't mean the older work is made irrelevant (or unimportant for audiences to know about). However, you should revisit your talks periodically (before you give them, of course) to determine what is still "fresh," and what is already well-known enough to gloss over in less detail.

As for assigning a "shelf life," that really depends a lot on your field. Some fields progress much faster than others, so a blanket statement such as "never talk about results that are more than a year old" is unlikely to be helpful.