The circles in Illustrator are flawed and here's why.
I discovered a major bug in Illustrator CC. The circles in Illustrator are not accurate. This is the case regardless of whether you use the ellipse tool or the arc tool. Make a circle. Check the dimensions to make sure it's accurate. Draw a radius line at 0º. Copy it 3 times and rotate the copies 90º,180º and 270º. When all of the copied lines are put on the center of the circle, they all touch the inner edge of the circle perfectly. However, Copy the 0º line and rotate it 45º. Put it on the center of the circle. It doesn't touch the inner edge of the circle. Copy the 0º line and rotate it 18º. Put it on the center of the circle. It’s even farther away from the inner edge of the circle. The circle is flawed.
Here is another way to see that the circles are not accurate. Make a hollow circle. Copy it and change the color of it's line. Rotate the copy 18º. Put it onto the original circle. They do not line up accurately. I spent an hour with an Illustrator expert while he had control of my computer through the support portal. He came to the same conclusion as i did. For some reason the circles in Illustrator are not perfect. He has spoken to his superiors and they are giving this issue priority. They were very happy that i told them about this and will hopefully solve the problem. For those of us that must have perfect geometry in our work, this is a very important issue.

AFvisions commented
I believe I'm noticing the same issue.
I have three rings of descending size and the two smaller ones should lay perfectly inside of it's larger parent ring. I had hoped using the pixel grid would fix the issue but I'm still unable to match the vector edges perfectly flush to the other. You'll notice in my image thin white lines where there are gaps between the rings. It's funny that the gap isn't all the way around, rather it becomes most apparent every 40 degrees.
The largest ring is 5.5" so the gaps are quite small, but as the design progresses it becomes more apparent.

Josh commented
My art is highly geometric and depends on perfect circles for perfect symmetry.
The imperfection of circles in Illustrator is highlighted when creating and rotating the "flower of life" design.
I am glad to see other people have noticed the same issue.
When working with a lot of circles, the imperfections/lopsidedness of them becomes very apparent with rotation of the work. My current piece should have perfect symmetry with every 60 degree rotation, but instead it stretches significantly (at least 5%).
I would also like to see an improvement to this flaw in the software.

Andrew B commented
By the way, the only 'pefrect' way I found to be working for me, is to use the Polygon tool with 360 'sides'.
The only downsides of it, is that it's still not a perfect cirlce, since it's still a polygon formed by 360 anchor points and when you duplicate and rotate it by 0.5° it still overlaps and won't fit perfectly, plus, it overloads the system and slows the computer while you use and adjust several similar shapes at a time. 
Andrew B commented
Until today this issue still exists.
It's so frustrating to work with several circles in Illustrator and see how the lines just won't align around the artboard, especially when I try to make geometrical shapes, for example: the flower of life, as a background for some artwork...
All anchor points are placed fine, but it just misalign due to the imperfect circles of the software...I really hope this thing will be fixed soon, as it affects lots of my works and designs.

Josh Balsillie commented
Anyone have a workaround to creating a perfect circle?
Equation for circle.
(x−h)² + (y−k)² = r²Is there any way to leverage the actual math for a circle to draw a circle?

Anonymous commented
same here. every non90°rotation discovers the not perfect circle. will take a look at that dynamicshapes plugin, thank you Egor!

Uwe Laubender commented
FWIW: The same thing with InDesign.
You cannot draw a perfect circle there as well.
The difference is minimal, but it is there.Regards,
Uwe Laubender
( ACP ) 
You will never have in perfect with Bezier curves.
Ellipse is always an approximation.
Let me cite this magnificent article (https://pomax.github.io/bezierinfo/):“Circles and Bézier curves are very different beasts, and circles are infinitely easier to work with than Bézier curves. Their formula is much simpler, and they can be drawn more efficiently. But, sometimes you don't have the luxury of using circles, or ellipses, or arcs. Sometimes, all you have are Bézier curves. ... So how do you draw a circle, or an ellipse, or an arc?
You approximate.
We already know that Bézier curves cannot model all curves that we can think of, and this includes perfect circles, as well as ellipses, and their arc counterparts. However, we can certainly approximate them to a degree that is visually acceptable. Quadratic and cubic curves offer us different curvature control, so in order to approximate a circle we will first need to figure out what the error is if we try to approximate arcs of increasing degree with quadratic and cubic curves, and where the coordinates even lie.”
You can buy DynamicShapes plugin from AstuteGraphics — it allows you to create ellipses with more points — or use a script (http://shspage.com/aijs/en/#circle)