Are there theories about the end of the universe where time ends?
A description in which spacetime comes back to a situation like the south pole in your question is the possibility in general relativity called "Big Crunch". We don't think the universe will do this (the matter density is not high enough) but you asked if there were models which allowed that sort of thing. The Big Crunch is in some respects like spacetime curving round on itself like the south pole of a sphere, so that there is no longer any direction called "south". However, it does not necessarily follow that the question "what comes after?" loses its meaning in a Big Crunch. Rather, in the conditions of the crunch, when the scale factor of spacetime has become small and the density large, our understanding of physics runs out and we do not know how to predict or even describe what the conditions are. So we just don't know if the concept of "future" ceases to have validity. Maybe it does, maybe it does not.
More generally, the human race does not have the data to answer a question like "will future always exist?". I suspect we do not have the intellectual capacity either. We can propose theoretical models, but all will reach a regime where there is no longer good reason for confidence that they still apply.
Currently the model believed by many is that matter will first get captured into black holes and then these will evaporate and then everything just fades into an expanding dust, expanding forever and doing nothing. But who knows? This model might be profoundly wrong. You can also find speculations about new bursts of "life" (that is, new patches of spacetime and matter undergoing a fresh big-bang-like behaviour) going on and on forever. Or perhaps the universe as we know it is in a state called "false vacuum" and the whole thing can get transformed by something like a phase transition. Or perhaps another type of transformation can happen.
Part of the aim of this answer is to reduce the level of confidence that readers place in whatever was the latest description they read which pronounced with confidence on questions concerning an ultimate future.
From your question, I reckon you don't want a too-in-depth answer. Please correct me if you want some equations to make sense of what I write.
This is a diagram widely used in cosmology. Roughly, you can see $\Omega_M$ as the matter density, and $\Omega_\Lambda$ as the Dark Energy density. There are various theories that try to explain what the universe might look like in the future.
The theory you were talking about (I presume) is what is commonly know as "Big Rip", and is represented by the dashed line on top. The universe never slows down its expansion, and eventually every particle will become a separate system unable to interact with every other one. In some sense, you can say that "there is no future", but please try not to use these kinds of expression while talking physics, as they can be really opinion-based.
The other probably interesting theory for your question is the dash-dot line at the bottom: it describes a "Big Crunch" and while this currently doesn't seem to have a lot of interest, it surely did when physicists were trying to figure out various models. If the matter density $\Omega_M$ is big, greater than one, the universe should slow down and eventually start contracting back until everything becomes really close. In some sense, this can also be seen as a "no-future scenario".
Lastly, you could just dive into a (non-rotating) black hole. You should reach its center in a finite amount of time and (while probably dying) you can't do anything else, ever, from there. So, again, no-future scenario.
There are many theories about the nature of Time. Whether the Universe is "open" with Time indefinitely long or "closed" with a finite Time ending in a Big Crunch is a matter of ongoing debate, although the standard model of cosmology has no mechanism for ending Time.
Some cosmologists propose a "Big Rip", in which the expansion of the Universe gets out of control and it destroys itself, like a balloon being over-inflated. Space and Time are the fabric of the balloon, so it is actually they which explode into nothing.
Some cosmologists propose a cyclical "Big Bounce", in which the Universe rebounds from the Big Crunch in a new Big Bang, endlessly.
These models are all based on the idea of Time as a single dimension conforming to particular solutions, or families of solutions, of Einstein's equations of General Relativity. But maybe Time itself is more complicated than that.
Stephen Hawking proposed a mathematical model of a complex Time which leads to a closed Universe where one dimension slides around the poles like just keeping going and suddenly you find yourself heading back towards the equator, so this dimension of time is finite but unbounded. Meanwhile the time which we perceive may still be open-ended.
Roger Penrose, in some ways Hawking's heir, has proposed a cyclic cosmology in with the Universe is constantly expanding but our measuring yardsticks for space and time get periodically reset so it looks like a bounce but isn't really.
If the Multiverse is a reality, does each Universe have its own independent flow of time or do they all tick in sync?
Many other fringe ideas have been investigated.
The big problem is finding experimental evidence to distinguish between these possibilities. It is a vastly difficult task. Some has been gathered, but it all remains either inconclusive or controversial.