I am referring to phrases such as: "Do you like her, or do you like like her." Can someone provide an explanation of this? There are many more examples but none come to mind at the moment.
The doubling of the word implies that both the reader and writer (or speaker and listener) understand that there are two different meanings for the word in question. It's something that is far more often used in speech than writing as it's possible to put a much more subtle emphasis on each word. In the case you mention, of course, the first meaning of like is "do you like her as a person" and the second "do you find her attractive". Other examples that spring to mind (in addition to the hot (temperature) and hot (spicy) that Stuart mentioned) are funny (either amusing or strange), see (literally or in a romantic sense) and the verb "to take out" (to take on a date or to kill/execute). The latter was used to great comic effect in the film "Pulp Fiction" where John Travolta's character is asked to "take out" the boss' wife.
This is known as epizeuxis (also called diacope), a process of joining words in immediate succession for vehemence or emphasis.
In the examples provided by the wiki link, it was used by William Shakespeare. Whether it dates back prior to that, for example, Homer is unknown. Being a Greek word, you would think so but, I would have to check my literature.
I'm going for antanaclasis
Antanaclasis is when a single word is repeated multiple times, but each time with a different meaning
Some examples similar to the manner you're using it in
"If you don't get it, you don't get it." —The Washington Post slogan
"Time flies like an arrow. Fruit flies like a banana." - attributed to Groucho Marx