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You can have a tangent line for every point on a circle, so the answer is theoretically infinite.

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Q: How many lines are tangent to a circle?
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How many lines can be drawn through a given point on a circle that is tangent to the circle?

Infinite lines because a circle has infinite lines of symmetry.


What is a theorem about two lines tangent to a circle at the endpoints of a diameter?

Two lines tangent to a circle at the endpoints of its diameter are parallel. See related link for proof.


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A line that is tangent to two circles?

... touches each circle in exactly one point on each circle. given any two circles where none is entirely inside or inside and tangent to the other, there are at most four straight lines that are tangent to both circles.


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