Here is the full list of all 23 Random War achievements.
Received "S" rank
Received "A" rank
Collected the first item
Recovered 100% of HP
You tried your luck.
You made an enemy ice cream.
You hit 3 enemies with a disc
You killed a giant fly
Defeated 100 enemies
Welcome.
Killed at least 4 enemies with a blast
Raised the level of an gadget.
You shoot a charged shot
Inferno Tank
Hit 4 or more enemies with a lightning bolt.
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Sorry...
Upgraded damage 3 times.
Upgraded Dash 3 times.
Hit 5 or more enemies with a nova blast
First play in surrounded mode
Wave 10
Wave 20