How many stars are there in the universe?

Enough that no one's ever going to actually count them to get an accurate census.

In 2003 a group of Australian astronomers (unfortunately, I'm getting this from BBC News and they neglected to mention specifically who) measured the brightness of the galaxies in one patch of sky, used that to estimate the number of stars they contained, and then used THAT to estimate the number of stars in the observable universe. The figure they came up with is about 7x10^22. (Roughly 0.1 mole, if you like).

Pieter van Dokkum of Yale University and coworkers revisited this in 2010 and found that there were probably more low-mass (and dim) red dwarf stars than was previously believed, which would make the number higher ... about 3x10^23 (roughly 0.5 mole).

I've seen other (more back-of-the-envelope and rougher) calculations suggesting the number could be as high as 10^24 (roughly 2 moles).

That's the observable universe. We don't (and probably never will) really know how if there's more to the universe than the part we can see, though some theories suggest it's at minimum anywhere from much bigger (250 times larger) to very very much bigger (>10^23 times larger), and it could be infinite. It's even possible that the universe is smaller than the observable universe, though this probably isn't terribly likely.

However, since the part of the universe that's outside the observable universe can by definition never interact with us in any way, we can ignore that and just say the number of stars in the part of the universe that actually matters is most likely 10^23, within an order of magnitude or so.

That is unknown, since the size of the Universe is unknown. In the so-called "observable Universe", it is estimated that there are about 170 billion galaxies; assuming that each galaxy has 100 billion stars, that would be in the order of 10 to the power 22 stars. However, the entire Universe is probably at least 1000 times bigger (in each of the three dimensions of space), and it may even be infinite, but this isn't currently known.

I estimate 100 E24 stars, 100 trillion Trillion stars..

There are over 10 billion trillion stars (100 billion+ per galaxy) in the universe.