Are there any anime series with neither an opening or ending song? I have watched quite a lot of anime over the last several years. And I noticed that I haven't seen a single anime which lacked both an opening and an ending song. There are some anime that don't have ED songs or don't have OP songs. 

But are there any anime that have neither of these?
Depends what you're looking for. Besides credits and providing a transition into the show, OPs and EDs also serve the purpose of presenting the sponsor messages when the show airs on real TV. These are always present on TV in some form.

That being said, if you grab a BD or a DVD of a show, the sponsors/credits are removed and you can skip the intros and have just the show. I know that there are also some shorts that aired online that didn't have any songs, but it's a small minority and things that most people never hear about.

So no, they don't really exist since they're necessary on television for financial and legal reasons.