To store information, you need a flash or EEPROM and, these memories have standard protocols (SPI or I2C). A microcontroller can communicate with these memories easily.
To implement these protocols with sequential or combinational logic, I think, it's impossible.
You can find controllers as cheap as $1 microcontroller - Google Search
There are single wire output ICs(with has start bit and stop bit usually this is used in sensor technology ), but I have not find any IC which can store the data and outputs serially ..
You can use a dip switch with a shift register then instead of burning the data every time just set the dip switch.If you are prepared to burn any chip that you want to change the data etc. then use apic 10f or 12f series that are less than 1$.
---------- Post added at 09:49 ---------- Previous post was at 09:07 ----------
You can use 1-wire devices it all depends on the protocol you want to use or just the raw 8 or 16 bits on/off **broken link removed**
You can buy a small avr (attiny12 or attiny13) for around $1 in QTY 10 or more. No other components needed except ONE attiny mcu.
WHY would you waste time on a series of switches or shift registers or anything else that will cost the same or more?