Frame dragging -- is there a "non-tiny" example? Now.  As I understand it, in fact, the earth (10^25 kg) creates a very small, very tiny, frame dragging effect. Indeed, we have measured this using satellite experiments.
So, the Earth (10^25 kg) creates a very tiny, minuscule, frame dragging effect.
I ask -- is there anything big enough, that it creates a let's say "non-tiny" frame dragging effect?
So for example, you astronomers who work with (say) super-clusters of galaxies, do you have to, as a matter of course, figure "frame dragging" when doing calculations regarding the size/shape/etc of these mega-objects?
Finally, I've tried to read what I can about the frame dragging effect of black holes - but I find it confusing and sort of not applicable to what I'm asking here.
Again to recap what I'm asking:
So, I know the Earth has a (tiny, minuscule) frame-dragging effect.
What about a star like the sun?  A galaxy?  A super-cluster?
How big does an object have to be to have "non-tiny" frame dragging effects?
Are frame-dragging effects an every day calculation for those who deal with (say) galaxies?  clusters?
Thanks!  I hope this is clear.

Note! here are some actual answers, thank to the amazing J.R.:
The sun and Jupiter - both still have extremely tiny frame dragging effects (about 100 times bigger than the Earth's extremely tiny frame dragging effect).
Galaxies - in fact - and I couldn't find this answer anywhere on the internet - galaxies in fact have trivial / basically zero frame dragging effect.  (Since they are so thin.)  Astounding!

Note! I previously included the following prelude to this question: If you spin a bucket of water, of course, the water "forms a concave shape" - Newton's Bucket thought experiment. As I understand it, physicists now believe that if you spin an astronomically large bucket of water then in fact very surprisingly ... it does not make the concave shape ... due to "frame dragging" cancelling the inertial force.  However, it now appears I was totally confused on this issue, so I have deletd the prelude to avoid confusion.  Sorry. (It's very much worth noting that all the discussion you can google up about "newton's bucket + frame dragging" -- seems very confused, so research with care on this topic!)
 A: The spacetime outside a spinning mass is described by the Kerr metric. To explain how the Kerr metric produces frame dragging is hard, because it's not something for which there's an easy intuitive model. Frame dragging arises because the spacetime geometry links the angle measured around the spinning object to time, and this means the angle changes with time. Points initially at some fixed angle get dragged in the direction of rotation.
The magnitude of the frame dragging effect is calculated from the Kerr metric, but it's not simply a case of how massive the object is. All rotating black holes contain a region called the ergosphere within which the frame dragging effect is so strong that nothing can resist it. The more massive a black hole is the bigger will be its ergosphere, but even tiny black holes still have an ergosphere.
The frame dragging angular velocity in the equatorial plane at a distance $r$ is given by (this equation is in the Wikipedia article I linked):
$$ \Omega = \frac{r_s\alpha c}{r^3 + \alpha^2 (r + r_s)} $$
where $\alpha$ is related to the angular momentum:
$$ \alpha = \frac{J}{Mc} $$
and $r_s$ is the Schwarzschild radius:
$$ r_s = \frac{2GM}{c^2} $$
The frame dragging gets larger as the distance $r$ gets smaller, but there's obviously a minumum value of $r$ that corresponds to the radius of the object. For the Earth you can't have $r < 6378$km because that's what the radius of the Earth is. You ask how the frame dragging changes with the size of the object: if you're thinking about an astronomical object like a star then as the mass increases it's size also increases, so the calculation isn't a trivial one. Plus big objects like stars are a lot less dense than small objects like the Earth so the relationship between mass and radius is different.
So how the frame dragging at the surface of a star compares to the frame dragging at the surface of the Earth depends on various different factors. However frame dragging is normally a very small effect for everything except superdense objects like neutron stars and of course black holes. It's the density that is the biggest factor.
Lastly, I've never come across the idea that water in a large enough bucket won't form a concave surface. Can you provide a link to whatever article it was you read that in?
A: Just to add to John Rennie's answer, the objects where we expect to see the largest frame dragging effects are spinning black holes.  There, there is actually a surface called the ergosphere (outside of the event horizon), where it is impossible for observers to stay stationary with respect to observers far from the black hole.  In a sense, their reference frame is being dragged faster than the speed of light.
This is interesting because there is a technique called the Penrose process where it is possible to rob angular momentum from the black hole while inside of the erogsphere, and convert that to energy that can escape the system.  In principle, this could be used to get more energy than would even be possible with nuclear reactions.  Of course, you need a spinnning black hole first...
