How would I make a keyboard shortcut to *maximize* the current window?

One of the easiest ways to do this is to use BetterTouchTool. It will let you bind a keyboard shortcut or a trackpad shortcut to many commands, including maximize window.

Once you are in the BTT settings page, do the following:

1. Go to the Keyboard tab on top.
2. On the bottom near Keyboard shortcut, enter in whichever keyboard shortcut you want.
3. For Predefined action, choose Maximize Window.

There are many other options related to maximizing besides "Maximize Window":

• Maximize Window Left
• Maximize Window Right
• Maximize Window To Bottom Half
• Maximize Window To Top Half
• Maximize To Next Monitor

I have assigned this script to ⌘M with FastScripts:

try
    tell application "Finder" to bounds of window of desktop
    tell application (path to frontmost application as text)
        set bounds of window 1 to result
    end tell
on error
    try
        tell application "System Events" to tell (process 1 where it is frontmost)
            click (button 1 of window 1 where subrole is "AXZoomButton")
        end tell
    end try
end try

It doesn't work with multiple displays though.

Applications that haven't been mentioned by other answers:

• ShiftIt
• SizeUp
• Slate

You can use an application like Divvy to do this. Divvy lets you select a rectangular area that you want the current window to take up. Selecting the entire screen lets you do a Windows-like "maximize" to have the frontmost window take up all available space on the screen.

It also lets you define keyboard shortcuts for specific rectangular selections that you want to use frequently. These can either be per-app, or a "global shortcut" that will work with all apps. To define them, right-click on the Divvy menu bar icon, select preferences, and select "Shortcuts". Click "new" and you'll be presented with a screen to define the selection and the keys you want to use:

BetterSnapTool (by the same author as BetterTouchTool) will allow you to do this. You can either drag the window to the top of the screen to maxmimise it (á la Windows 7) or define a shortcut key. There are also lots of other options such as maximise left-half or right-half etc. Its also only £1.49/$1.99 on the app store, which is almost free :-)