Collection of interesting integer sequences whose terms are still being discovered
I think showing high-school students such sequences is useful, because it demonstrates that all of mathematics is not done and dusted. We have far more questions than answers, which should be somewhat obvious, as every time you engage with a difficult problem you're going to generate lots of questions, only a few of which you'll have the energy, or interest, to answer. All of the sequences I've listed so far have Numberphile videos to accompany them. You can show students these sequences by simply showing them each Numberphile video. However, many more sequences can be found via the OEIS. Look for the keyword "hard."