Alright so, as mentioned, I'll share a bit about each Astrocade game that mine came with. Remember: I have no real nostalgia for this thing.
We'll start with the four built into the system: Gun Fight, Checkmate, Calculator... and Scribbling...
Gun Fight is a home version of the spectacular black and white arcade game and it lives up to the name well enough. The game shows off all the aspects of the Astrocade's unique controller: moving your character with the joystick, aiming your gun with the paddle, and of course firing with the trigger. If you have a friend nearby, this is maybe one of the best games on the system, in my opinion.
Checkmate is a variant of Blockade (sort of similar to a multiplayer Snake game for the younger crowd) which also plays pretty dang well. It's also one of the few games I have that has AI so I could actually enjoy it for an extended period of time.
Calculator is...a calculator. Remember that back in the day systems like this were often marketed as home computers so having something like this built in makes sense. I'm not really sure how practical it might have been but this is the mid to late 70s we're talking about.
Finally, Scribbling...took a few resets to actually do anything. Eventually I was able to draw some crap on the screen, change colors, etc. Then I turned it off again.