I can see two paths.
First, there's Roger Penrose's Conformal Cyclic Cosmology in which the beginning of our Universe is the end of the previous one.

This presents a picture of an eternal Universe which goes through creation-expansion-destruction epochs similar to how cyclical time is described in South Asian philosophy.

The second will take a some explanation.
- Premise 1: Our experience of reality is that logic exists. For example, causal statements depend on the existence of the arrow of time.
- Premise 2: In any given infinite set of chaos, there will always be at least one subset of order. For example, within an inifinite set of monkeys banging on typewriters the complete works of Shakespeare will be written by happenstance.
- Premise 3: Logic is a kind of order.
- Premise 5: We call experiential reality "The Universe."
- Conclusion: The Universe is a logical subset of a larger alogical (this is a neologism, not a typo) reality—where the term "alogical" represents all possible logics and all possible nonsense.

This presents a picture of a Universe with a beginning which was rooted in the ordering of chaos similar to several ancient cosmogonies. (Note that this is not *ex nihilo*, from nothing, but from something we can't comprehend.)


So, It's certainly possible that the Universe had a beginning, but that its predecessor didn't (or cannot) as well as the possibility that the Universe has always existed and will always exist in some epoch or another and that our Universe is but one of these.
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