Start with a triangle, and construct its inscribed circle. Take the three points where the inscribed circle is tangent to the triangle, and construct a new triangle with those points as the vertices. Then construct the inscribed circle of this new triangle. Continuing this construction indefinitely gives us a sequence of inscribed circles and triangles that telescope down to a limit point. This is just the construction from this question, but we start with a triangle instead of a circle (and so no choice of points is being made).
In the comments Steven Stadnicki brought up a good question: does this construction telescope down to a defined center of the original triangle? By "defined center", I mean any of the many many points that have been given a name like prefixcenter of a triangle. It's naturally not the incenter, since the limit point will literally be the limit point of the incenters of the sequence of nested triangles, which isn't constant. It can't be the orthocenter or circumcenter since those could easily lie outside the triangle, and I checked that it's not the centroid.
I suppose an equivalent question would be: does there exist an alternative way to construct this limit point using only finitely many steps that maybe someone has considered before?
Not a complete answer, but the first inner triangle is called the intouch triangle or contact triangle of the large triangle; some more information is at Wolfram MathWorld here, where it is mentioned that the triangles in this procedure approach equilateral ones.
The Encylopedia of Triangle Centers may be useful here, though I didn't see anything relevant in the first 1000 by searching for the phrase "intouch". For what it's worth, a friend of mine knowledgeable about triangle centers didn't know of a common term, and expected that the resulting center wouldn't be very "nice" - not having barycentric coordinates expressible as a polynomial in terms of the side lengths or angles (or trig functions thereof) of the original triangle, for instance.