Sword Art Online

Where there is Log Horizon, there is SAO; that's just how it is. Though, I'm putting it in here for a slightly different reason.

SAO and Log Horizon are similar, but focus on different aspects. SAO has the "game" feel that Log Horizon does, but takes it to the next level based on the fact of death being absolute. SAO enacts the drama side of being trapped in another world really well, which is why I think you'll like it.

Of course, there is more to SAO than that, but I want to focus sorely on the first season (Aincrad). We get more of the awesome fights that are far and few between in Log Horizon; mainly, boss raids. Though SAO does not go into as much detail as Log Horizon on these raids, the fight scenes are still pretty cool.

I'd recommend you checking out SAO, at least the first season, if you are wanting a more drama out of the show, and don't mind being stuck to the main character (unlike LH where we follow many characters).