The main candidates are series with short episodes (e.g. ONAs, anime that air in short timeslots, etc.), since the openings and endings are more likely to be cut due to time constraints. But ack, most of them still have openings and endings to some extent!
For example, consider Tonari no Seki-kun: It's 21 episodes, 7 minutes 40 seconds each. But despite the time constraints, it still has an opening and ending! They're shorter than usual, mind you, at 45 seconds each. But they're still proper songs, so this doesn't count.
Consider also Mahou Shoujo Nante Mou Ii Desukara: It's 24 episodes (2 seasons), 4 minutes each. Despite the super short length, this series still has a proper opening song that lasts 45 seconds. It doesn't have an ending song or credits. But it has the opening, so this doesn't count either.
I know of some more likely candidates, but I'm not sure if some of them completely fit the bill. But anyway, I'll list them:
Rockman.EXE BEAST+: An anime series consisting of 26 episodes, 10 minutes each. Each episode has an opening sequence that lasts for 20 seconds, but it's just credits text flying at you in a Windows Music Visualizer-esque cyber tunnel thing, set to an instrumental version of one of the soundtracks from the series. Not really a proper opening song, in my view. This series doesn't have any ending credits or songs either. (It does have, however, a 15-second preview of the next episode.)
Melancholy of Haruhi-chan Suzumiya: An ONA consisting of 26 episodes, around 2~5 minutes each. Well, it has an "intro song" of 1 minute, but this intro song is kept separate from all of the other episodes. For each episode, the "opening" is basically a sequence of Haruhi smashing into the screen, which lasts 10 seconds. The episodes have ending credits, but they only last for 15 seconds, and it's white text on a black background with some light instrumental music. (Reminds me of Western cartoons...)
Nyoron Churuya-san: An ONA consisting of 13 episodes, around 2 minutes each. It doesn't have any opening song or credits. The episodes do have ending credits, but they only last for 15 seconds, and it's just white text on a white background, set to some recorder music and the graphic of a single slowly spinning smoked cheese.
Barakamon: Mijikamon: An ONA consisting of 13 episodes, around 2 minutes each. The episodes don't have either an opening or an ending song. The only "opening", so to speak, is a 6-second segment where Handa and Naru say "Mijikamon" in front of the title card.
Tanaka-kun wa Kyou mo Kedaruge: An ONA consisting of 35 episodes, ~30 seconds each. The only "ending" to speak of is a 2-second static title card at the end with some credits displayed.
Side note: Though most anime out there tend to have at least one of an opening or an ending, for the series that air in shorter timeslots, I've seen them occasionally cut the opening/ending song short due to time constraints. So perhaps you could say that for those specific episodes, the anime has neither an opening nor an ending song.
- For example, Ryuusei no Rockman consists of 55 episodes, 10 minutes each. It has an opening sequence with a proper song that lasts 60 seconds. It has no ending song or credits. However, in episodes 48, 49, 54, and 55, the anime truncates the opening at 6 seconds (just long enough to show the title). The credit texts continue showing up for the first minute or so into the episode, but the opening song isn't played.
Well, that's all I got.