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.
The obvious answer is hydrogen and helium plasma but the nuclear fusion can also create heavier elements. Are these heavier elements a significant portion of the core?
As said in dmckee's answer, no, the core of the Sun is much too cool (about ~15 000 000 K) to burn any other than hydrogen into helium. The triple-alpha process, which converts helium into carbon, only kicks in somewhere around 80 000 000 K, depending on density.
That said, the CNO cycle does modify the internal abundances very slightly. That is, the CNO cycle is a catalytic process, so the abundances of those elements are driven to the values at which the reaction proceeds at an equilibrium rate. Glancing at a solar model I have lying around, this leads to about a 10% enhancement of the central nitrogen abundance, and a corresponding decrease of the carbon and nitrogen abundances.
Do the heavier elements "sink" to the "bottom" of the core, like iron has during planetary formation?
Actually, yes, they do! We refer to these processes as atomic diffusion. The one you're thinking of is known as gravitational settling. In short, yes, heavier elements "sink" towards the centre. This process takes a long time to make a meaningful difference: billions of years. For the metals, it isn't important, but it is actually important for the helium abundances. There was a minor revolution in the mid-1990's when this effect was included for the first time, and it led to a much better fit of solar models with respect to helioseismic observations.
Presumably, during the Sun's formation it would have accreted heavy elements made by previous generations of stars - does this just get added to the mix?
You're quite right: the Sun's initial composition reflected that of the nebula from which it was born, itself a product of whatever star(s) preceded it. The primordial mixture is expected to be fully mixed before a star starts burning hydrogen into helium. The reason is that the star (or, at least, our models) goes through a phase where the whole star is convective, so everything gets churned up and homogenized.
Best Answer
The Sun didn’t “select out” heavy elements from the cloud. The planets selected out light elements because they don’t have enough gravity to hold on to their hydrogen and helium.
Source: http://curious.astro.cornell.edu/our-solar-system/53-our-solar-system/the-sun/composition/201-does-the-sun-have-any-heavy-elements-beginner