I am almost certain that time travel in loops is impossible, and I am deeply saddened by this. I am a time travel fan, and there would be no causality ping pong for me. Ever. 
On the other hand, I think we can cheat to allow for partial time travel to alternative timelines.
So here's a thought experiment: It should be theoretically possible to reconstruct the past (or a past if it turns out the universe is non-deterministic going backwards) on a computer, using the present as a bootstrap. Then, provided you could upload your brain to it, you could just go there. If this were to happen: Would you be able to distinguish it in any meaningful way from actually travelling there? Or, if the simulation is perfect, would the ensuing events in the simulated timeline differ from the ones after a hypothetical time travel? If the answer to either of those is no or too hard to determine, I would consider those to be the same thing. More or less.
Of course, that's impossible, because it's not only computationally intractable, but would require a prohibitively accurate and wide numerical snapshot of the present. However, it is a comfort to consider that an imprint of our past is preserved, even if can't use it in that manner, and that we can access a substantial part of it through history book produced using far less fancy methods.
In a way, the past is like another universe to us, except it is within our reach in a way no other universe would ever be – we can actually observe it, and that's awesome. That's oughta be enough time travel for anybody, why can't we be ever satisfied?
Unanswerable mind puzzle: If humans didn't have a writing system, and somehow miraculously survived without one until the disappearance of the rest of the galaxies in our expanding universe, would future scientists be able to unequivocally predict their existence from the surviving geological and archaeological record?

We haven't observed anything that requires time travel for its explanation, so the proposition is a useless complication of the universe. Furthermore, it would also be detrimental to what we have observed, because there's the grandmother of all grandfather paradoxes – if we will one day time travel, how come we didn't already prevent our own evolution from happening?
Technically, like all grandfather paradoxes, that's not really a problem – something will happen to prevent this impossible chain of events from occurring, thus there would be some bizarre limits on what could happen, but I am pretty sure that this "something" would amount to one of: 1) humans not existing; 2) time travel not possible. Since I have a suspicion 1) isn't true, I'm going for 2).
That doesn't mean that it's not an option at all, but with little to support it, it's nearly as wishful as hoping magic incantations were real. Sub-atomic processes don't count, since they do not allow for the backwards transfer of information – and backwards transfer of information is what it is all about. Some funny effects that seem to, to our limited perception of reality, propagate back in time, cannot justify something that would fundamentally alter how the macroscopic world functions.
Technically, like all grandfather paradoxes, that's not really a problem – something will happen to prevent this impossible chain of events from occurring, thus there would be some bizarre limits on what could happen, but I am pretty sure that this "something" would amount to one of: 1) humans not existing; 2) time travel not possible. Since I have a suspicion 1) isn't true, I'm going for 2).
That doesn't mean that it's not an option at all, but with little to support it, it's nearly as wishful as hoping magic incantations were real. Sub-atomic processes don't count, since they do not allow for the backwards transfer of information – and backwards transfer of information is what it is all about. Some funny effects that seem to, to our limited perception of reality, propagate back in time, cannot justify something that would fundamentally alter how the macroscopic world functions.
On the other hand, I think we can cheat to allow for partial time travel to alternative timelines.
So here's a thought experiment: It should be theoretically possible to reconstruct the past (or a past if it turns out the universe is non-deterministic going backwards) on a computer, using the present as a bootstrap. Then, provided you could upload your brain to it, you could just go there. If this were to happen: Would you be able to distinguish it in any meaningful way from actually travelling there? Or, if the simulation is perfect, would the ensuing events in the simulated timeline differ from the ones after a hypothetical time travel? If the answer to either of those is no or too hard to determine, I would consider those to be the same thing. More or less.
Of course, that's impossible, because it's not only computationally intractable, but would require a prohibitively accurate and wide numerical snapshot of the present. However, it is a comfort to consider that an imprint of our past is preserved, even if can't use it in that manner, and that we can access a substantial part of it through history book produced using far less fancy methods.

In a way, the past is like another universe to us, except it is within our reach in a way no other universe would ever be – we can actually observe it, and that's awesome. That's oughta be enough time travel for anybody, why can't we be ever satisfied?

Unanswerable mind puzzle: If humans didn't have a writing system, and somehow miraculously survived without one until the disappearance of the rest of the galaxies in our expanding universe, would future scientists be able to unequivocally predict their existence from the surviving geological and archaeological record?
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