That's what appears to be on offer from Physicists Paul Steinhardt and Neil Turok and is beautifully described for the layman right here.
I'm particularly intrigued by this passage:
"One can further show that, as the cycles proceed, the relaxation slows more and more as the cosmological constant gets smaller, so exponentially more time is spent in cycles with a small cosmological constant. In this picture, it is natural to expect the tiny value for the constant we observe today."
Surely this implies that - within the context of their conjecture - we could work out how many cycles we must have gone through already, in order for the cosmological constant to have reached its present minute value. And doesn't that imply a true "beginning" - albeit gazillions of times further back than we thought? A beginning at which the cosmo constant was the current predicted value 10^100 times what it is today. Or not?
It would be interesting to know just how many hundreds of trillions of years or cycles that implies.
It's also important to note that the cyclic nature proposed by this conjecture does not in any sense imply "repetition". If we don't survive the collapse of this Universe, we don't return in the next. The universe which arises in any given cycle will, inevitably, have a slightly different set of physical laws to its predecessor and thus be dramatically different in its history and appearance.
As a consequence, this model also provides an alternative to the "multiverse" as a solution to the anthropic problem. In fact, this conjecture is about a different sort of multiverse. It's a serial multiverse, as opposed to the more widely known parallel multiverse.
I think this idea might have legs...