Why are you using a SR to drive a display, and not a display driver IC? Will this be a multiplexed or direct drive display? Does it have to be one or the other?
If using 74hc595 cascaded, you will need 7x 74hc595, 7x single digit displays, 7 bypass caps, 49 current limiting resistors, = 63 components
You could use 7x 74hc7447, 7x single digit displays, 4x 74hc595, 13 bypass caps, 49 resistors = 65 components
You cannot use multi-digit displays unless you create a multiplexing circuit, which can be done with combinations of 74hx595, 74hx7447, transistors, resistors, capacitors, and the Arduino must control the multiplexing, making your sketch much more complex and difficult. = ??? components
But, i have thousands of resistors, capacitors, 50 or more 7 digits displays, and 15 74HC595 (my father has a lot of elements, also an analogic osciloscope, why send more money buying more components if I have lots?
Once all its mounted, it will work, I am very good at programing, coding will not be a problem.
I have tin, a lot of PCB boards, and, in summer, a lot of free time xD
Yes, your proposition is legit, efficient, and maybe the best option, but, having lots of components, makes me want to use them instead of buying more.
3x 74hc595
3x 0.1uF bypass caps
21x series resistors for led segments, e.g. 510R
3x series resistors for transistor bases, e.g. 220R
3x npn transistors, e.g. bc337
9x CC single digit displays
3x Arduino digital outputs, ideally MOSI, CLK, SS for h/w SPI
Each '595 drives 7 segments of a display, so 3 displays are lit at once. Assuming you don't need the decimal points, the remaining '595 outputs on the 3 chips drive the base pins of 3 transistors. Each transistor sinks current from the common cathode of 3 displays. This way you can drive 9 displays with a 1-in-3 multiplex, keeping them quite bright.
If you can spare 3 more Arduino pins, use these to drive the transistors. You can then lower the segment series resistors to 330R for a little more brightness.
In any design you should avoid overloading the '595 with too much current (max 70mA per chip).
EDIT: I seem to have increased your 7 digits to 9 digits above!