I have been seeking out Tokyo's best bars long enough to have accumulated a fair-sized database on the subject. Even so, for the best part of a decade, a search with the keywords safe + sane + fun + Roppongi have turned up zero results. By safe I mean you don't have to watch your purse. By sane I mean you can talk to the person next to you without shouting. By fun I mean the person next to you would be someone you would want to talk to. And by Roppongi I mean within the first few blocks that hug the four corners of the crossing.
With the discovery of Kun-pu, I have finally found my haven. (Though next time I might also factor in cheap, which Kun-pu isn't; the seating charge at the bar is ¥800 per head.)
This is an establishment that spans two floors, with a restaurant on the second floor and a bar on the third. The former is exempt from the charge and always full, thanks to the chef's flair in the kitchen. But the latter, even with the charge, is well patronized till sunrise.
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