Okay, now I understand better. With that in mind, we can go with the 299s you have already.
Let me explain one shift register first then we can connect the remaining shift registers up.
You hold the right serial input (DSR, pin 11) to a logic 1, since we are shifting a 1.
Since we are going to shift that 1 from left to right, you hold mode input S1 (pin 19) to a logic 0 (GND) and mode input S0 (pin 1) to a logic 1 (+5v).
The OE1 and OE2 inputs (pins 2 and 3) are held fixed at logic 0 so that the outputs (I/O 0 thru I/O 7) are always enabled.
Since the outputs of the 299 can drive the LEDs directly, connect the anode of each LED to each of the 299 outputs through a 120 ohm resistor. The cathodes of the LEDs are all connected to GND.
You bring the MR (pin 9) input low (logic 0) and then high (logic 1) whenever you want to set all the outputs to the logic zero state, turning off all the LEDs.
Now, every time the CP input transitions from low to high, the outputs of the shift register will fill up with 1s from left to right. Each CP input transition will add another 1 to the outputs, turning ON another LED.
To add another shift register, connect the I/O7 output of the first shift register to the DSR input the second shift register, and so on to add more shift registers. Connect all the MR inputs together. Connect all the CP inputs together.
Am I making sense?