Best way to control multiple LEDs?

I wanted to build a project that would have 4 output to create chasing LEDs effect

1 1 0 0
0 1 1 0
0 0 1 1
1 0 0 1

The output is the easy part (a pair of flip flop circuits can do it) but most IC do not take kindly to driving 100 LEDs per pin.

Available source is 2.5v and I plan to use white 3mm LEDs. How can I interface multiple LEDs to the chip so the current draw won't fry the chip? Also at about 2.5v (less any voltage drop if I use silicon switching device) I won't need resistor. Is there any drawback to having 100 LEDs in parallel? If one LED is going to suffer thermal runaway due to defect, it'd just burn itself out then the rest of LED would continue as normal.

I could use higher voltage and resistor but I am lousy with SMD and wiring regular 1/8w resistor in a small project is going to be a messy rat nest of nightmare. (not to mention resistor + heatshrink tubes are thicker and would require redesigning) so I am hoping to just skip the resistor.

TIA for any suggestion!

4 outputs

That binary sequence is the goal? 4 bit patterns looping? So many questions...

What components do you have?

Yes, that's the output sequence in loop to make LED light seems to chase. Right now I have a 555 timer and a 4013 flip flop IC but I can use Arduino instead.

These chips are designed to run as low as 3v but I've had them running stable at lower voltage when the circuit is running on slow clock (4 or 5 Hz)

I was thinking, how about a 4017, each count turns on a seperate transistor which powers the leds in the first sequence. Next pin turns on the next transistor which turns on the 2nd set.

You can have up to 10 binary patterns then.

I am concerned with the lack of current regulation for the amount if leds you have, a higher voltage and place the leds in series with a resistor on each string, but with such a low voltage.. meh

With only 2.5 volts, white LED:s needs 3-3.5 volts + current limit resistor.
I think if Will be a little bit dark. Or have I missed something?