Here are some of the board games I have referenced or referred to, and those that have come up in conversations and discussions.
City building and building board games
AlhambraThis is a great game that has many levels of play. Using a range of playing strategies such as collecting palace tiles, collecting resources, organising your turns or plays and keeping an eye on your opponent, makes for a great and complex game without the need to keep score or writing anything down. The game rewards efficient play by giving more turns to any player with the exact amount of resources when purchasing a palace tile. Scoring occurs in three parts, at roughly even times throughout the play. There is also a nice twist in creating walls in the palace as only the player with the longest wall gets points. This can often be the deciding factor for a winner. It is best played with three or more players.
Strategy war board game
This is also a great game, and there are a variety of ways of playing the game. This is a far simpler war game then Axis and Allies, where players are given a set number of army tiles that they place strategically or randomly on the map of the world (depending on the rules). Each player takes turns in collecting and placing troops, attacking other territories, and then finishing their turn by re-arranging one set of their troops from one territory to another. Winning a territory allows the player to pick up a card, which can be traded for armies in future turns. Combat is a simple dice roll, where attacking armies must roll higher than defending armies. You can choose the number of armies to attack by using a maximum of three, and you must always leave at least one army in every territory you control. It is best played with three or more players.