There has long been a line of speculation as to "what happened before the Big Bang?; Could there have been many Big Bangs?" This is an area of almost pure speculation. We simply do not have theories which are robust enough to believe at those scales. However let me mention some of the speculations. In Einstein's theory of General Relativity (GR) there are some very powerful theorems, primarily due to Penrose and Hawking and Penrose, called the singularity theorems. These state that given certain assumptions about the behaviour of matter (that the energy of matter as seen in certain frames is positive) then singularities in the past in our universe are inevitable. Thus, in order to evade these theorems and get rid of the "Big Bang" singularity, one must either believe that General Relativity breaks down, or that these energy assumptions are wrong. Now, there is an example in GR of a spacetime which does not have an initial singularity but rather has a universe which "bounces." That is the DeSitter universe, in which one adds a cosmological constant to the theory (which is effectively adding something with a very very large negative pressure to the theory which can evade some of the energy conditions.) In some ways the Hartle Hawking prescription for the "wave function of the universe" is precisely placing a constraint on the conditions that the universe must bounce, and evades that initial singularity.
Linde, in his eternal inflation scenarios, assumes that the universe is, over most of its size, dominated by such an effective cosmological constant, and occasionally due to some chance fluctuation in the matter, the universe becomes dominated by the ordinary sorts of matter and then produces a universe like ours. But most of the universe (and the past of our part of the universe) is exponentially expanding. (He wrote a Scientific American article on this about two years ago). Others like Peebles have also speculated on a progressive series of such big bangs, big crunches and successive big bangs and wondered about the role that the second law of thermodynamics would play in such a scenario. Note that one would expect that the universe at its end time would be very very different from its beginning (it would be highly inhomogenoeus and clumpy with much of the matter having produced singularities in black holes, etc.) and it is hard to see how such an inhomogeneous universe could then have reversed itself and started to expand again. This has led some (eg. Wheeler, Hartle) to speculate that perhaps there is some law that states that the universe returns to its homogenous state at the end again ("terminal guided statistics" where one places not only initial but also final conditions on the evolution of the universe). This leads to weird speculations about the effects that such terminal conditions would have, for example, on life (does one grow younger as the universe collapses back to a small size?) and other dynamic processes. All of this has the sword of experiment hanging over it since it seems that the universe actually does not have enough matter in it to cause it to eventually re-collapse, but will rather expand forever, and thus could not recycle itself.
Many physicists tend to dislike the thought of a beginning. And there do appear to be highly speculative but not absurd ways of getting around the necessity for the universe to have started from an initial singularity and grown from there. However, I would say that none of these has any real support either in the details of theory or in experiment. The most conservative state is still that the universe began about 10-20 billion years ago in a singularity - that there was a single Big Bang which began everything. On the other hand speculation is fun.