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As is well known, a triangle center can be exterior to the triangle. So, just how far from the triangle can one of its centers be?

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    Do you mean the center of the circumcircle? – Arthur Nov 27 '18 at 18:19
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    If you scale such a triangle up, you can make it as far as you like ;) – angryavian Nov 27 '18 at 18:20
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    The question isn't clear. How are you defining the distance? Euler tells us that the distance between the incenter and the circumcenter satisfies $d^2=R(R-2r)$. Is that the sort of thing you are after? – lulu Nov 27 '18 at 18:23

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As far as you want it to be. If you have a triangle whose circumcenter is 1 unit away from the triangle, you can just double all the sides of that triangle, and you get a triangle whose circumcenter is 2 units away. And so on.

Even if you limit yourself to, say, triangles with no sides longer than $1$ in order to make this scaling infeasible, you can do it another way. Consider if you have an isosceles triangle where two sides are $0.5$ and the angle between them is close to $180^\circ$. The closer to $180^\circ$ you make that angle (while keeping the triangle isosceles and the two side lengths constant), the further away the circumcenter moves.

And there is no bound on how far away the circumcenter can move in this manner; as our triangle comes closer and closer to being a degenerate triangle (three points on a line), the circumcircle comes closer and closer to being that line (which in this case is what a circle with infinite radius looks like).

Arthur
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