Does future diary have any fighting in?
It has fighting and romance.
I'll try not to spoil anything in the information I give you.
The main female protagonist is madly in love with the main character (there's your romance).
The fighting is when everyone is hunting each other trying to become God.
Hopefully you don't get confused by the end of the anime, it does get a bit weird.
Watch SAO (1st season is best season), Blue exorcist (great story),Full Metal Alchemist (I watched the 2004 original) , Deadmam wonderland (Short and anything but sweet).
SAO is okay. If you think about it, they basically repeat the plotline for S1P1 and S1P2. In addition, the incest and attempted r*** were unnecessary imo. If you're watching SAO for anything, watch it for the battles, not the plot.
*I do ship them, don't hate meh*
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DanMachi - The main character does have a crush on someone, but they don't go too far into a relationship. The plot is centered around a RPG with a leveling system and statistical abilities. The characters don't fight each other, as they fight monsters (mobs) more often. If you're fine with a PvE(Entity) type anime, this is great!
You want fighting animes without romance ;-;
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I can't really connect to your anime choices very well because I don't like those type of animes. I feel they've repeated the same kind of storyline over and over, which is why they have so many episodes. As to why they have so many fans, maybe you can clue me in on it.
