Are Four 74HC595 Sift-registers in a row too many?

Try this PaulJakob:

/* code for sending data to 4 shift register which are daisy chained together
 MOSI goes to Serial In on chip 1 (pin 14), Q7/ (pin 9) goes to Serial In on chip 2, etc. 
 SCK goes to ShiftClock on all 4 devices (pin 11)
 ss goes to RegisterClock on all 4 devices (pin 12)
 SerialClear (pin 10) is pulled high on all 4 devices
 Output Enable (pin 13) is pulled low on all 4 devices
 0.1uF capacitor from each chip's Vcc to GND
 */
 
// used Sketch:Import Library:SPI to make this show up
#include <SPI.h>

int ss = 10;
int i = 0;
int shiftreg = 0;
byte byte1 = 1;
byte byte2 = 1;
byte byte3 = 1;
byte byte4 = 1;


void setup() {

  pinMode (ss, OUTPUT);
  SPI.begin();  // library takes care of D11 MOSI, D12 MISO, D13 SCK

}

void loop(){
  // loop thru 8 times, walk a 1 across all 4 registers
  for(i = 0; i < 8; i=i+1){
    digitalWrite (ss, LOW);
    SPI.transfer (byte1);
    SPI.transfer (byte2);
    SPI.transfer (byte3);
    SPI.transfer (byte4);
    digitalWrite (ss, HIGH); // all outputs change on ss going low to high
    delay(1000);
    byte1 = byte1 <<1; // update the data
    byte2 = byte2 <<1;
    byte3 = byte3 <<1;
    byte4 = byte4 <<1;
  }
  // after all 8 outputs are high one time, reset for the next pass
  byte1 = 1; // update the data
  byte2 = 1;
  byte3 = 1;
  byte4 = 1;  
}