It is less than a quarter of an hour, or a full quarter of an hour, or what amount of time is being talked about here?
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足足有一刻钟 here means "a full quarter of an hour".
足足 means fully. The word is usually emotional, implying that that amount is a big deal. E.g. 我在这里足足等了你两个小时. (implying: two hours is really long time. It's a complaining about the fact that you kept me waiting for that long.)
PS. Compare 足足有两小时 and 足有两个小时. They mean the same(a full two hours), but the former puts an emphasis. 足足 sounds like almost exceeding the amount(or perhaps already exceeds a little bit?).
It is (at least) a full quarter of an hour. Note that 足足 is used for emphasizing that.
◎ 足足 zúzú