Dominion. The strategy arises from observing how the specific cards that are part of any given game interact with each other.
Agricola. There's a lot of "designed" complexity here too, but a good deal more arises, particularly when you look at planning the critical eight first actions, and the path to the first additional family member.
"Codenames" is a clever word game. It has a downside that it requires at least 4 players. There is a board of words (each word is on a double-sided card and there are a few hudreds). One person on each team has a diagram with certain words (array indices) he must communicate to another team members. He does that by saying a word (word association) and a number, which means how many words he's communicating. For example, if there's a ring, a journey, and a monster you could say "Hobbit 3". Unlike a similar game, Dixit, this one has better longevity as it doesn't promote saying very obscure things only one person in group will know.
Neuroshima Hex! is a bit nerdy looking, but it's simple without expansions. Retro apocalyptic setting with mutants and evil robots. A knife fight in a telephone booth, plays well up to 4 players. You fill up the hexagonal board with your troops (they mostly don't move) rotating them so that they will hit the most enemies. Then it either fills up completely or one of players plays Battle token, and everything goes off like a chain reaction. Units hit in melee or shoot in order of their initiative. High interaction, tight mechanics. Most people prefer it as a duel game though. You can play it online too.
Lords of Scotland is a fiendishly tricky game that appears simple at first. It's very quick to teach, but there is both strategic and tactical aspect to it and many non-obvious paths to victory. Beware of high player interaction (some people like it, some are hurt). It's rather chaotic in higher player kinds, you need to appear relatively harmless to win.
Tigris and Euphrates is an old classic that has very few tile types to place, but they result in complex outcomes. I can't say very much about it because I only played it once on a game show, but it's on my to-buy list.
I cannot recommend Tigris and Euphrates enough. It used to be at the top of BoardGameGeek in the old days. Some of my friends likes to sell it to newbies as 'its like chess, but for four person'. By the way, author of this game (Knizia) is a master Board game builder and has created many more games worth looking into.
Dominion. The strategy arises from observing how the specific cards that are part of any given game interact with each other.
Agricola. There's a lot of "designed" complexity here too, but a good deal more arises, particularly when you look at planning the critical eight first actions, and the path to the first additional family member.