Before anime got what little respect it enjoys today, it was very much of a secret thing. Fan tapes passed hand to hand. Bits and pieces seen at conventions. It was very much a process of discovery. There was no context given for what we were seeing. The translations were inconsistent, and if you did find something dub it was often very off model (one famous example was giving Captain Harlock a John Wayne voice!)
I remember that one of the very first anime badass I saw was Cobra. The scene was perfect overkill. Our hero was riding a futuristic motorcycle and being chased by a ship easily as big as a Star War's Star Destroyer. He turns around and then something happens to his arm. It becomes a gun. Suddenly there is a cut to the ship and this one poor red shirt is wetting himself and he just has enought time to say.. "PSYCHOGUN!!" Then Cobra shoots the damn ship out of the sky with one shot.
Add an obnoxious personality that somehow was a babe magnet, his own personal spaceship, and his android Lady and you had Cobra who was sort of an anime version of Han Solo with a much bigger gun. I was amused years later to see the animation show up in the video Girlfriend.
Hyper Police was also another early favorite. It was like a weird version of the Gamma World game. Mutants and other creatures were the majority and justice was handled by various bounty firms. Our heroes are an unlikely team of a cat girl and a fox spirit. It just is a super wacky combination of gritty action and cute. Yet it worked for some reason.
One of the few times anime broke near the mainstream in the early days was on the budding Science Fiction Network (before it became Syfy) Even though I had seen it before I was thrilled when "Tank Police" was shown on TV hosted by Ralph Bakshi! It was something of a vindication that we weren't just being weird.
Tank Police was also something of an odd mixture of cute and gritty. Set in the future things were so bad that the cops now ran around in super large tanks. The plots were incidental to massive property destruction.
Of course besides mindless violence fans were attracted by the infamous Puma sisters. No doubt early viewings of the Pumas did a lot to encourage the Furries. Hey no one is perfect. My only question was, where did they hide those guns?
Of course you can't think of the early days without thinking of Lum. The most out right silly thing one is likely to see, it was like I dream of Jeanie on acid. That we didn't get 90% of the jokes based on Japanese culture didn't stop it from being screamingly funny.