What would you call someone who imposes on other people's generosity?

A scrounger:

a person who borrows from or lives off others

You could also call that person a sponge:

sponge n.
8. One who habitually depends on others for one's own maintenance.

The term sponger is also used to mean the same thing, chiefly in British English.

We used the term "mooch" (def 7) when I was younger. I think that one literary offering is "takers" (Ayn Rand's Atlas Shrugged)

Would you allow the Yiddish schnorrer?

“One who habitually takes advantage of others' generosity, often through an air of entitlement.”

A leech

3 : a hanger-on who seeks advantage or gain

[Merriam-Webster]

"Opportunist":

One who takes advantage of any opportunity to achieve an end, often with no regard for principles or consequences.

From the FreeDictionary.com

A taker. It's colloquial, but spot on.

A freeloader

a person who habitually depends on the charity of others for food, shelter, etc.

[The Free Dictionary]

One meaning of the word parasite is:

One who habitually takes advantage of the generosity of others without making any useful return.

Exploitative, as in 'tending to make use of selfishly or unethically'.

I'd agree with Dan; I'd call that person a mooch. From NOAD:

mooch
(verb) ask for or obtain (something) without paying for it
(noun) a beggar or scrounger

You can also use moocher in place of mooch. Wordnik defines moocher as "A person having a tendency to repeatedly ask help of others, especially if they are making little effort to help themselves."

In British English, you could also use cadge or cadger.

User was the first that came to mind... That person is such a user. I've also heard cat. ...Once you give, they keep on coming back.

I call them Shameless.

They have no shame and will beg for scraps.