Retina, 31 bytes

.+ $* Mr!&`11+ m`^(1+)\1+$ $1 1 

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How it works

The regex /^(1+)\1+$/ captures the largest proper divisor of a certain number represented in unary.

Retina, 31 24 bytes

7 bytes thanks to Martin Ender.

.+ $* M!&`(1+)(?=\1+$) 1 

Try it online!

How it works

The regex /^(1+)\1+$/ captures the largest proper divisor of a certain number represented in unary. In the code, the \1+ is turned to a lookahead syntax.