For me it is the fact that our blood contains iron. I earlier used to believe the word stood for some 'organic element' since I couldn't accept we had metal flowing through our supposed carbon-based bodies, till I realized that is where the taste and smell of blood comes from.
It's even crazier because you don't need to reach the speed of light. It'll happen in a smaller degree for any speed. Even in mundane conditions.
For example, if your twin spent four days in a 300km/h bullet train, for you it would be four days plus a second.
Usually this difference is negligible, but for satellites (that run at rather high speeds, for a lot of time, and require precision), if you don't take time dilation into account they misbehave.
(For anyone wanting to mess with the maths, the formula is Δt' = Δt / √[1 - v²/c²]. Δt = variation of time for the observer (you), Δt' = variation of time for the moving entity (your twin), v = the moving entity's speed, c = speed of light. Just make sure that "v" and "c" use the same units.)
Yes I knew about that and I'm glad that doesn't make it crazier for me, instead it makes it easier to accept. If it were something that happened only after hitting some arbitrary speed value I'd be a lot more mentally damaged
To be fair the only ones that don't get mentally damaged at all with this stuff are theoretical physicists. After all being crazy makes you immune to further madness.