Some anime are just absolutely bewildering. Whether it's an abstract anime that's rife with symbolism, like Mawaru-Penguindrum, or a meta anime that breaks the fourth wall, like FLCL, these confusing anime are likely to leave you scratching your head.
Note that confusing doesn't automatically imply a lower degree of quality. In fact, some of the best anime ever made, like Neon Genesis Evangelion and FLCL, are difficult to understand without background knowledge and careful thought. These series are still worth watching, but it's best to do so with the understanding that they might not make sense on the first viewing, if at all.
These series are not just simple anime to relax with and have on in the background. These are challenging and complex programs that demand total attention and commitment.
FLCL is one of the trippiest anime ever made, so while it's enjoyable whether you understand it or not, it not an easy watch. On a thematic level, it's a coming-of-age story, but on a literal level, it's about a pink-haired alien who opens up a portal in a 12-year-old's forehead and pulls robots out of it. Teasing out the actual meaning of this utterly insane anime is easier said than done.
Neon Genesis Evangelion is so difficult to understand that fans have written entire essays in attempts to break down all its content. Not only does it demand viewers follow the conflict between NERV and the Angels, it also brings in Freudian concepts, religious allusions, and complex metaphors.
If someone is looking for a straightforward mecha show, this is not that, but if they feel like giving their brain a workout, they should watch Evangelion.
Serial Experiments Lain is notorious for not making sense. When 14-year-old Lain receives an email from a seemingly deceased classmate, she discovers the Wired, a combination of the internet and virtual reality. As she spends more time in this mysterious world, the boundaries between the Wired and her actual life become porous and blurry. Lain questions the nature of reality, but it never provides its viewers with any clear answers.
Monogatari is a dense series of several anime, all centering around the nearly immortal Koyomi Araragi and the various supernatural happenings that surround him. Many of the basic conflicts that Araragi deals with are easily comprehensible, as he's trying to help protect people from curses, hexes, and other magical problems.
However, the details surrounding them are not simple at all. Every image, every word spoken, every piece of text scrolling across the screen is of some metaphorical significance. It takes multiple viewings to understand it all, and even then, it's a challenge.