From Förberedande kurs i matematik 1

If we complete the square,

\displaystyle x^{2}+x+1=\Bigl(x+\frac{1}{2}\Bigr)^{2}-\Bigl(\frac{1}{2} \Bigr)^{2}+1 = \Bigl(x+\frac{1}{2}\Bigr)^{2} + \frac{3}{4}\,,

we see on the right-hand side that we can make the expression arbitrarily large simply by choosing \displaystyle x+\tfrac{1}{2} sufficiently large. Hence, there is no maximum value.