What does the word "dog-boy" mean in the context below?
Do you know how uncivilized Leonard was when I took him in?
No.
Oh, it took me forever to get him on a bathroom schedule.
He would just go whenever the mood struck him.
Like a dog-boy.
Exactly.
(The Big Bang Theory)

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Answer

Sasha:

Clearly "dog boy" is a confused reference to a dog that urinates   and so forth  anywhere he wants.    However,  I have lived in the USA all my life and never heard the expression.

If you ever  have a conversation with me, and use the phrase "dog boy" I won't know what you are talking about, and neither will the rest  of the population.

That's the problem with attempts to learn something termed "English" from modern TV  Situation Comedies.   There is no absolute meaning to the phrase,   "dog boy" although it is clearly a colloquialism  the script writers for the TV program "The Big Bang Theory"  have  introduced.  It means about as much as using an expression like "Cow Child" or   "Cat Kid".  ---Warm Regards, Bruce

Answer

Like Bruce says, "dog boy" is a term the writers made up for the show. It refers to a dog that urinates anywhere it wants and a boy that is not yet potty-trained.