People don't tend to find that unintuitive. The Greeks had a nice geometric example in a square of side length 1:

    ┌┬┬─┬───┬───────┐
    ├┘│3│   │       │
    ├─┘-│   │       │
    │ 64│   │       │
    ├───┘   │       │
    │       │       │
    │ 3/16  │       │
    │       │       │
    ├───────┘       │
    │               │
    │               │
    │               │
    │       3/4     │
    │               │
    │               │
    │               │
    └───────────────┘

3/4 + 3/16 + 3/64 + 3/256 + ... is easy to visualize as successively filling in three quarters of an ever-smaller residual square. Intuitively, no matter how finely you detail it, you're never going to stop fitting inside the original area-1 square.

edit: better text art

Now squeeze the top-left piece so it's half as wide but twice as tall (i.e. the same height as the full rectangle), and do this recursively. Same area, right? Then stack the L shapes on top of each other. Then you have one of these horns.

Huh? The horn is a three-dimensional object; the square exists in 2-space. You can't make the horn from pieces of the square even if you allow deformation.

I don't think that's paradoxical. We deal with convergent infinite series all the time in everyday life.

10/3(3.333...) is 3 + 3/10 + 3/100/ + 3/1000 + 3/10000...

π(3.1415...) is 3 + 1/10 + 4/100 + 1/1000 + 5/10000...

Its quite easy to see that both of these series will be finite, and many numbers, lets say 4 or 3 + 5/10 will be greater than either of them.

Or in general, any infinite size n-dimensional structure can be embedded in some finite size n+1 dimensional structure.

This sum's finiteness seems pretty intuitive:

    9 + 0.9 + 0.09 + 0.09 + 0.009...

Good example. Minor nitpick, you listed 0.09 twice.