We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, \({\mathcal L}(x,\del_x)=a_{ij}(x)\del_i\del_i\), for \(n\geq 3\). We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed \(y\), we construct a solution of \({\mathcal L}Z_y(x)=0\) for \(0<|x-y|<\e\) with explicit leading order term which is \(O(|x-y|^{2-n}e^{I(x,y)})\) as \(x\to y\), where \(I(x,y)\) is given by an integral and plays an important role for the fundamental solution: if \(I(x,y)\) approaches a finite limit as \(x\to y\), then we can solve \({\mathcal L}(x,\del_x)F(x,y)=\de(x-y)\), and \(F(x,y)\) is asymptotic as \(x\to y\) to the fundamental solution for the constant coefficient operator \({\mathcal L}(y,\del_x)\). On the other hand, if \(I(x,y)\to -\infty\) as \(x\to y\) then the solution \(Z_y(x)\) violates the "extended maximum principle" of Gilbarg & Serrin \cite{GS} and is a distributional solution of \({\mathcal L}(x,\del_x)Z_y(x)=0\) for \(|x-y|<\e\) although \(Z_y\) is not even bounded as \(x\to y\).