When IO is limited on a uC, how do you move the logic away from the uC?
Bruno gave a good answer, but I'd like to make a few notes.  

The infamous 70 mA
The 74HC595 is often used as I/O extended, and as often used to control a series of LEDs. What is often overlooked is that you may operate it out of spec this way. The datasheet says that the total supply current shouldn't be more than 70 mA, Absolute Maximum Ratings (AMR), so you better stay away from that by some margin. Therefore 8 LEDs at 10 mA is too much, and at 20 mA you'll exceed AMR by no less than 130 %! The limit is probably due to the current capacity of the bonding wires, and then the too high current may not just deteriorate the part's performance, but permanently put it out of order if that wire would break.  

But last night I woke up because I had An Idea. The 70 mA limit goes for both Icc and ground current, then why not split our total or 80 mA and let Icc have half of it and ground current the other half? All you have to do is reference 4 of the LEDs to ground (active high), and the other 4 to Vcc (active low). Then the current of the former comes from Icc, the current of the others goes to ground. You'll be able to use 15 mA LEDs this way. (It's so simple that I feel like an idiot not to have thought of it sooner.)  

The combined clocks
I thought Bruno saved an extra I/O pin by combining the shift register's clock with the latch's. Seems I misinterpreted his answer. I still want to expand on this option.  

What happens then? The table on page 5 of the datasheet says:  


  contents of shift register shifted through; previous contents of the
  shift register is transferred to the storage register and the parallel
  output stages  


(highlighting by me)
So it's not the new data but the previous which is being latched. Not a real problem, just make sure to shift in an extra dummy bit to latch the last data, or everything will be one pin wrong.

Combining the clocks also means that the outputs will toggle all the time while you're shifting new data in. The latch's function was actually to avoid that. In many cases this won't be a problem, if you can do it fast, but in worst case situations you may get undesired effects. Worst case could be using the 74HC595 to multiplex a display at a very high scan frequency + have a very long chain of shift registers + have all 1's except for one 0, a LED that's off + a dark room. Since that LED sees all the 1's passing every so often in a dark room it may light very faintly.  

Or if you combine that high frequency multiplexing with a relay control output. All zeros and then a 1 for the relay may mean that the relay's output is not long enough high to pull-in.

Granted, those are extreme cases, but I would keep the serial and latch clock separate whenever possible if you want to multiplex, or otherwise have high update rates.