Another game that popped into my head today - Ide Yousuke Meijin no Jissen Mahjong 2
Its was a Famicom Mahjong game released by Capcom (obviously only in Japan). Comes with its own special controller as well to make it easier to play (to quickly select tiles etc.)
What sets this one apart from other Mahjong games I've played is that it has its own dedicated story/RPG mode, where all battles are Mahjong based. Its actually a really fun game, gets pretty darn hard late in the game as the AI gets better and better (you can get items that even the odds though). Its one of Capcom's weirder titles that I ended up loving, obviously if you don't like Mahjong you'll never get anywhere in the game, but if you its well worth the import.
Sticking with Capcom, the Tenchi Wo Kurau series (both the Arcade and Famicom games) are another good shout. All the games are based on the manga of the same name - its basically yet another retelling of the 3 Kingdoms era of China (think Dynasty Warriors) told from Liu Bei's point of view, with his oath brothers Zhang Fei and Gwan Yu tagging along for fun.
The arcade games are scrolling brawlers, the first takes place entirely on horse back and is a pretty fun game, but maybe a bit too long. Port wise it also got a pretty good PC Engine port. But the sequel, while removing the dedicated horse back combat (its a powerup now) is better - more moves, better graphics while retaining a pretty tough difficulty curve, still feels a bit too long as well. Again, port wise the Saturn got a good (albeit expensive now) port.
The Famicom games though are turn based RPGs, again based around the same characters and story. What makes the battles fun though, aside from your normal attack, magic etc. attacks is the "all out" mode which is just a way to quickly have both sides fight until only one is left. Just makes those turn based battles that much faster. Otherwise they are fairly standard RPGs, reecruit more generals, beat up Lu Bu again and eventually slap some sense into Wei. Its just a strangely addictive duo of RPGs, and I'm not normally one for turn based RPGs but these just do it for me.