Eight coins for the fair king
Python 3, 113 62 bytes



for i in[1]*3:x|={a+b for a in x for b in x}
while{i+1}&amp;x:i+=1


Here x is the input as a set of ints, and i is the output.

Try it online!

(Thanks: Erik the Outgolfer, Mr. Xcoder, Lynn)