The Gradius series had Zubs that teleport onto the screen and will most likely kill you if you are not prepared for them, and some volcano stages have Erupting Rocks that will force you to the top or bottom of the screen.
The rocks in Gradius III AC's lava stage were more like Demonic Fireballs, they appear in large numbers, split into indestructible fragments when shot, and to make it worse, the passage gets narrow near the end of the level.
Touhou has Lily White. In the ninth game, Phantasmagoria of Flower View, she will randomly appear in both player's fields and rain down hell upon you for no reason other than she can. Sure, you can shoot her down, but she comes back a minute later. This is especially irritating when fighting Medicine or Eiki. Goddamned Lily.
Also, Orin, who makes her appearance in the 11th game, Subterranean Animism, as the Stage 4 mini-boss... twice (like Alice in Perfect Cherry Blossom), then as the Stage 5 mini-boss... then as the Stage 5 boss... and finally as the Stage 6 miniboss. I don't care what you have to say about Lily, at least she only stays for one wave. This bugger tails you through three stages, and only lets up when Hell's fires get too hot for even her.
Metal Slug has Helicopters that constantly hover overhead and fire on you. On their own, they're not too bad, but they have a habit of attacking you while being backed up by tanks, soldiers with knives, and other such ground infantry, making it very difficult to kill them off due to the wave of attacks coming from both ground and air. It's even worse when they come in a row and start circling overhead, constantly raining down bullets everywhere. In those cases, at least, shooting the leader causes a chain reaction to take out the rest, giving a nice score bonus.
You know what's worse than the helicopters? The normal soldiers that throw fireworks at you. Despite having the same amount of health as other soldiers, the player can hit their projectiles, not only blocking your shots but also making the objects much more unpredictable to dodge. Add in the usual threats(including the helicopters) and you're all but guaranteed to die many times.
Spheres Of Chaos has black holes. Those which attract objects aren't too bad, but those which repel, are, making objects to go fly everywhere and player has to fly very close to them to shoot them due to their repelling force. This is especially bad with multiple black holes.
They also have Dodgers which are normal enemies with the ability to dodge player's fire, often making them hard to catch.
Fortunately, the Probes can be easily beaten for easy money if you have a Spathi Eluder, one of the best ships in the game.
The worst place to encounter them just HAS to be in Ur-Quan space. Especially the border. It is entirely likely you will be stuck in an impossible-to-escape series of sequential battles as more Ur-Quan and Slylandro continuously spawn.
During combat in any of the three installments in the series, asteroids play a similar role, straddling the line between a mindless environmental hazard and an active adversary. They are spawned constantly, most of them along random trajectories, but some are intentionally hurled at your ship to make things harder for you. They'll block your shots, push you into an enemy's line of fire, ruin your perfect maneuvers, and in the worst case push you straight into the planet (for massive damage).
R-Type players hate Cancers with a passion. Why? Though extremely easy to kill (one shot from anything makes them dead), they home in on you while firing shots with near-perfect accuracy, and often (this is the kicker) come in on the left side of the screen. Unless you are powered-up, you cannot fire in that direction. Use your imagination...
Space Invaders Extreme has Reflector Aliens, which bounce your shots off, Assault Aliens, which take two hits to destroy, the first making them fly down to kill you, Flipping Aliens, which turn on their sides to become harder to hit, and much more.
Grakin Assault Helicopters and Mokum Assault Helicopters from Heavy Weapon. The former enters screen, and then it spams homing missiles on your tank. These helis have quite a bit of health, and they tend to come with other enemies, making them annoying to defeat. As for the latter? They're like the former, only with more health, and twice the missile spam. Gah.
S-34 Reflex fighters. Like the helicopters, they spam homing missiles but thankfully leave the screen. They're annoying is because they sport Deflector Shields that absorb your regular shots (given that your regular weapon is a rapid fire Spread Shot, it happens a lot). When this happens it sprays out unkillable purple plasma shots in completely random directions while alsomaking a rather irritating sound when they do so. It's a good thing said shields do not absorb your special weapons fire.
Raptor: Call of the Shadows has Mooks, both airborne and grounded, fire orange homing balls at you to make things more difficult. Individually, the shots are weak, but in numbers they can whittle down your shields and armor if you're lazy enough to not get out of the way. A large number of these Mooks will use these on you in the later levels and harder difficulties. Coupled with the tendency of Mooks to appear suddenly on the screen and blast you with more dangerous weaponry, and the homing balls are more likely to hit you since you're too busy avoiding to get hit by other Mooks while simultaneously trying hard enough to pick them apart for cold, hard cash to spend at the Emporium.