How to calculate currents and voltages with transistors circuit
You are forgetting that you can overdrive a base so that it goes into saturation. This is desirable to get the transistor fully turned on.

Your calculation for I7 is correct:

$$ I_7 = \frac {V}{R_3} = \frac {5 - 0.6}{25k} = 0.176~mA $$

You then calculate the maximum collector current using \$ I_8 = \beta I_7 \$
Lets say \$ \beta \$ = 100, then \$ I_8 = 0.176 * 100 = 17.6~mA \$.

Now look at R1. The maximum current through it is given by 

$$ I_1 = \frac {V}{R_1} = \frac {5 - 0.6}{1k} = 4.4~mA $$

Clearly the maximum possible value for \$ I_8 \$ is this 4.4 mA + 0.176 mA from \$ I_7 \$.

The calculations indicate that we could reduce the base current into Q1. In practice we want to ensure that the drive is adequate to cover variations in \$ \beta \$ due to production spread or even transistor substitution. Drive it hard and make sure it's fully 'on' is the normal approach.