A worn out string looks very frayed or falling apart.
I like to play mine till that happens.
Or if I get a knot that I can’t get out. 
For me it depends on the string. Dragon strings get better with age and the other toxics last forever.
I like my kitty strings to be pretty fresh, though they definitely last a long time.
Twisted strings I find are somewhere in the middle depending in type. d last forever and maintain their feel. TypeA1 I use when I want a really stiff string, so maybe a week or so before I change it when it gets more flexible. Type B holds its feel for quite a while.
Stringlab are also in the middle, they do keep the same feel for quite a while after breaking in.
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