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#C (gcc), (削除) 91 (削除ここまで) bytes

j;m;t;f(x){for(j=m=0;t=x>>j+8;m<t?m=t:j++)t=t<<j|x%(1<<j);return m;}

-23 bytes from Colera Su

Supports up to 2**31-1

Try it online!

Starts with the low 8 bits (j=0), then goes up, changing output if the number with bits [j,j+8) cut out is bigger than our current, and continuing until x has no bits above j+8