How many times has the matter in our Solar System been recycled from previous stars? I've got a basic understanding of these facts:

*

*The Universe is a little over 13 billion years old.  Our Galaxy is almost that old.

*Our Solar System is roughly 4.6 billion years old.

*The heavier elements (carbon, oxygen, etc...) are only produced by stars.

When our Solar System coalesced, the matter that makes up the planets came from a previous generation of stars.  (Meanwhile our Sun is busy making its own slightly-heavier elements, but those aren't ours.  Those belong to the next "generation" after our star dies.)
So we're all sitting on a rock whose matter was spewed out of a star (or stars) that preceded this solar system. But what I want to know is:
How many cycles of coalescing, fusing, nucleosynthesis, exploding -- and then repeating the cycle -- have the atoms in this ball of rock I'm standing on been through?
How do you know this?  Can you tell by the make-up of the matter in the Solar System?  Looking around the Orion arm to see remnants of our former home?
 A: The "generations" of the stars are surely not as sharply separated as you suggest. It's true that the percentage of heavier elements - or "metallicity" - is increasing as the stars continue in their thermonuclear fusion. But most of the stuff that our Sun burns is still Hydrogen - and it's the same "initial generation" Hydrogen as any previous stars were using. Stars are being born continuously and the metallicity is usually a little bit higher than for the previous star.
In the era of "nucleosynthesis" that ended about 3 minutes after the Big Bang, mostly light elements were created in a thermal equilibrium at huge, "nuclear" temperatures that existed at those times. A good theory of nucleosynthesis predicts that most of the elements in the Universe should be hydrogen, with a smaller amount of helium and some trace amounts of lithium and other elements. The observations confirm that the predictions are pretty much accurate.
Heavier elements were created in previous stars. But it is not correct to suggest that the Sun is "almost entirely" built out of a recycled material. Quite on the contrary: it's more accurate to say that the Sun is mostly built from the hydrogen that was created in the first minutes after the Big Bang - and it is also "contaminated" by metals and other heavier elements from the previous stars. Those impurities are important for our life - and industry - but they're not important for the ability of the Sun to burn.
A: Its likely a complicated mix, possibly including some virgin material, clouds of which occasionaly are absorbed into the galaxy, and some material that may have multiple recyclings. And not all the material that goes into a star is reacted before its blown off, so some (maybe much) of the hydrogen in the sun could have once been in the outer layers of another star. And even some first generation starts (pop III) that were low mass still haven't completed their lifecycles, so it is just a chaotic mess of gas and dust that gets pushed around by gravity and stellar winds for billions of years, plus some new gas that has recently fallen into the Milky Way all mixed up, but not so well mixed up as to be of uniform composition.
A: The elements that make up the bulk of the Earth were part of the presolar nebula. A similar (though not identical) mixture of elements is found in meteoritic material, which is thought to more accurately represent the mean abundances of that nebula (minus the volatiles) and indeed also agrees with the abundance patterns in the Sun.
There are grains of material trapped inside these meteorites that consist of solids that were already present in the presolar material. These are important because these grains were thought to have formed in individual stellar events and their isotopic compositions can be studied. These tell us that the Sun formed from material that has been inside many different stars of different types.
Stellar evolution and nucleosynthesis calculations tell us the same story. For example, whilst most of our oxygen was made in massive stars that underwent a core collapse supernova, such events do not produce much carbon. The C/O ratio tells us that most of our carbon comes via the winds from intermediate mass AGB stars. Heavy elements like Uranium are dominantly produced in supernovae (though current thinking is that merging neutron star binaries may be major contributors), but others like barium are not.
The details of how many generations have preceded the Sun and Earth has no single answer. Much of (perhaps 90%+) the solar hydrogen and helium could be pristine; some will have been through more than one star. Heavier elements (bar some lithium) will have been through at least one star. The fact that we have s-process elements like Ba, La and Ce, which are formed by neutron capture onto iron-peak elements, tells us those have been through at least two stars. 
However, these are vast understimates. Mixing in the interstellar medium is reasonably effective. The material spewed out from supernovae and stellar winds 5-12 billion years ago has had plenty of time to mix throughout the Galaxy before the Sun's birth. Turbulence and shear instabilities should distribute material on galactic length scales in a billion years or less (Roy & Kunth 1995; de Avillez & Mac Low 2003), though local inhomogeneities associated with nearby recent events can persist over $10^{8}$ years. If this is the case, then the Sun is the product of the $\sim$ billion stars that died before it was born.
A: There are three general populations of stars, based on their metallicity, which is the abundance of heavy elements on their atmosphere. That abundance is an indicator of the abundance of heavy elements of the initial cloud that gave birth to the star. 
So we have Population I stars that are metal-rich, Population II that are metal-poor and Population III that are metal-free. Population III stars are the first stars that formed in our universe and they must have been very massive and sort lived. Population II are the second generation of stars that created the heavier elements. Finally, Population I stars are the most recently created stars that also have heavier elements that were created in the Population II stars. 
Our Sun is a Population I star, so in a sense the Sun is a third generation Star. Taking into account the age of the universe and the average life-time of the more massive stars that form the heavier elements in supernovas, then the assumption that the Sun has used the material of at least two previous stars is reasonable.    
A: A very qualitative way to look at it:


*

*The Earth, and therefore you are formed of the same material that contributes to the metallicity of our Sun

*Our Sun is a population I star, which means that it has a relatively high metallicity indicative of having formed after the heavy and short lived stars of population II had already had their big blow offs.

*The population II stars divide into early and late groups, and all post-date the assumed population III stars.


From this I conclude that a non-trivial number of the nucleons in your body have been part of a few stars. Maybe as many as five. As Georg notes there has been time for the most prolific path to include many stars (dozens?).
Certainly all the carbon, nitrogen, oxygen and trace elements that make up your body have been part of at least one star.
None of these facts shed much light on the average star-membership-history of the nucleons that make up your body.
A: First of all -- check the metallicity, then the nucleocosmochronology. That gives the rough idea about the topic. And also that the topic is a currently developing area in cosmology. 
It seems that the concise and experimentally confirmed theory of formation of first stars as well as theory of reionization are required to answer your question. And there no such theory (yet).
