Can a disk of radius R be covered by two smaller disks of radius r1 and r2 ?

Pretty sure no, but proving it is more difficult.

No. Is there such a thing as proof by "just look at it and it's obvious"?

Let's deal with two circles describing the edges of the disks. There are two points at the intersections of the two circles. The two points can't be farther apart than the diameter of the smaller circle.

I'm pretty sure I could prove the rest, but I don't know how proofs are done and it would take time and diagrams and sloppy language. A short version where I don't back up what I'm saying:

Because the distance between the two points would be the diameter of the larger circle if the points were equally far apart along the circumference of the larger circle, and because the smaller circle must have a shorter diameter than the larger circle, the two points must describe both a larger and a smaller segment of the circumference of the larger circle. If you draw a line between the two points, halfway along that line will be a point on the line connecting the centers of the two circles. This line must connect to two points on the smaller circle, one outside the big circle and one inside the big circle. Because of this and the shorter length, the small circle represents the smaller segment of the large circle's circumference. Because small segment + large segment = circumference, the small segment must be less than half the circumference of the large circle.

Therefore, the segments covered by two small disks must be less than the total circumference of the large disk, and therefore two small disks can never completely cover the edge of the large disk.

Actually, proof is simple:

Consider 3 circles, A,B, and C. diameter d(A)>d(B) and d(A)>d(C). Overlap A with B. Since d(B)<d(A), there must be two points p1 and p2 on the circumference of A distance d(A) apart that are not covered by B. Intersect p1 with C. Because d(C)<d(A), p2 is still not covered.

fold the first circle in half. Voila.

Here is an additional proof.

This type of reasoning may be extended to R3 where three smaller spheres can cover a larger sphere in much the same way that three smaller disks can cover a larger disk.

WLOG assume the disk is centered on the origin O1(0,0). Again WLOG, assume a second point is centered on the point O2(x,0). The circle centered at O1 intersects the y-axis at two points whose distances from O2 exceeds the radius of the larger circle. There is no way that a third circle can contain these two points whose distance apart is the diameter of the large circle.