Sam accurately identified the two factors that let to your problem:
This is fairly simple, it's a result of how you duplicated the frames, it's also rotating off center.
a) Were you duplicating the original layer every time and then applying the rotation?
b) Or were you duplicating the previously duplicated and rotated image and then rotating it again?
What most people don't realize is that in a pixel based program such as PS (ie, in contrast to a vector based program such as Illustrator), every time you rotate, or even scale an image by some arbitrary amount, you lose some detail. This is most obvious at sharp edges, but it occurs throughout an image. If your next rotation step is based on the results of the previous rotation, the loss of detail adds up almost perfectly linearly (ie, not exponentially) with the number of steps.
For example, here is a starting image that consists of a 700px wide red rectangle on a black background. It has pixel-sharp edges.
Next, here is the result Sam's method (b) applied in a series of 5 degree steps, rotating the rectangle 360 degrees around its center:
As you can see, the edge became wavy and soft. If I had done two complete rotations instead of one, the waviness and softness would each have been twice as much.
I used the plain "bicubic" interpolation method option for rotations. If I had used "bicubic softer", there would have been less waviness but more softness. If I had used the nearest neighbor (NN) interpolation method, the edge would have remained pixel sharp, but there would have been little triangle-like serrations along the edges.
Also, the larger the pixel dimensions of the objects being rotated, the waviness / softness becomes smaller as a fraction of the total size of the object. Since you were working on an object with very small pixel dimensions, these effects were fractionally much larger in your case and caused the "ripples" in your spinner.
The reason that Sam asked about method (a) is that this is the preferable method. This way, each new layer would be only one rotation away from the original, and the errors would not be cumulative like they are in method (b).
Another method that would greatly reduce this problem (and still be very easy from the POV of convenient repetitive keystrokes) is to start with a much larger image, say photographic sized, ie, several thousand pixels on a side, do your rotations and any other transformations you need at that scale, and then, as the next to the last step before you make an animated GIF out of the layers, down-rez it to the final size you need for your spinner.
HTH,
Tom M
PS - BTW, I changed the title of this thread to reflect the fact that these types of errors are simply cumulative, not geometrically increasing, and certainly not exponentially increasing with the total number of steps.
PPS - If you are familiar with PS's vector tools, you can successively rotate shapes / paths without running into this problem because they are not based on the discrete nature of individual pixels.