It is known that digital logic can be implemented with 555 chips:

• Digital logic using 555 chips
• 555-logic: designing digital circuits with 555s only!
• Boolean Logic with 555s
• Concept: Four 555 timers into a gated D latch
• More 555 logic (plus 3 555 timers into a active-low-gated not-D latch)

Specifically, an A&!B gate (inhibit gate) can be constructed, which is functionally complete, and thus, 555 logic should be Turing-complete.

We can do a little better than building an entire machine out of inhibit gates, however. The 555 also contains an RS latch which comes in handy when implementing sequential logic.

There is also a long tradition of building clocks from (sometimes obscure) logic families. Some noteworthy examples:

• Transistor Clock and TTL Clock by KABtronics
  
• Neon Clock (and its sequel) by PA3FWM
• Diode Clock by Hackaday's own Ted Yapo

555 Time aims to follow in these footsteps by creating a clock entirely of 555 timer logic.