No Stupid Questions

(I'm going to take the question seriously)

Supposing that you're asking about a digital clock as a standalone appliance -- because doing the 69th second in software would be trivial, and doing it with an analog clock is nigh impossible -- I believe it can be done.

A run-of-the-mill digital clock uses what's known as a 7-segment display, one for each of the digits of the time. It's called 7-segment (or 7-seg) because there are seven distinct lines that can be lit up or darkened, which will write out a number between 0 to 9.

In this way, six 7seg displays and some commas are sufficient to build a digital clock. However, we need to carefully consider whether the 7seg displays have all seven segments. In some commercial applications, where it's known that some numbers will never appear, they will actually remove some segments, to save cost.

For example, in the typical American digital clock, the time is displayed in 12-hour time. This means the left digit of the hour will only ever be 0 or 1. So some cheap clocks will actually choose to build that digit using just 2 segments. When the hour is 10 or greater, those 2 segments can display the necessary!number 1. When the hour is less than 10, they just don't light up that digit at all. This also makes the clock incapable of 24-hour time.

Fortunately though, to implement your idea of the 69th second, we don't have this problem. Although it's true that the left digit of the seconds only goes from 0 to 5 inclusive, the fact remains that those digits do actually require all 7 segments of a 7seg display. So we can display a number six without issue.

Now, as for how to modify the digital clock circuitry, that's a bit harder but not impossible. The classic construction of a digital clock is as follows: the 60 Hz AC line frequency (or 50 Hz outside North America) is passed from the high-voltage circuitry to the low-voltage circuitry using an opto-isolator, which turns it into a square wave that oscillates 60 times per second.

Specifically, there are 120 transitions per second, with 60 of them being a low-to-high transition and the other 60 being a high-to-low transition. Let's say we only care about the low-to-high. We now send that signal to a counter circuit, which is very similar to a mechanical odometer. For every transition of the oscillating signal, the counter advances by one. The counter counts in binary, and has six bits, because our goal is to count up to 59, to know when a full second has elapsed. We pair the counter with an AND circuit, which is checking for when the counter has the value 111011 (that's to in decimal). If so, the AND will force the next value of the counter to 000000, and so this counter resets every 1 second. This counter will never actually register a value of 60, because it is cut off after 59.

Drawing from that AND circuit that triggers once per second, this new signal is a 1 Hz signal, also known as 1PPS (pulse per second). We can now feed this into another similar counter that resets at 59, which gives us a signal when a minute (60 seconds) has elapsed. And from that counter, we can feed it into yet another counter, for when 1 hour (60 minutes) has passed. And yet again, we can feed that too into a counter for either 12 hours or 24 hours.

In this way, the final three counters are recording the time in seconds, minutes, and hours, which is the whole point of a clock appliance. But these counters are in binary; how do we turn on the 7seg display to show the numbers? This final aspect is handled using dedicated chips for the task, known as 7seg drivers. Although the simplest chips will drive only a single digit, there are variants that handle two adjacent digits, which we will use. Such a chip accepts a 7 bit binary value and has a lookup table to display the correct pair of digits on the 7seg displays. Suppose the input is 0101010 (42 in decimal), then the driver will illuminate four segments on the left (to make the number 4) and five segments on the right (to make the number 5). Note that our counter is 6 bits but the driver accepts 7 bits; this is tolerable because the left-most bit is usually forced to always be zero (more on this later).

So that's how a simple digital clock works. Now we modify it for 69th second operation. The first issue is that our 6-bit counter for seconds will only go from 0-59 inclusive. We can fix this by replacing it with a 7 bit counter, and then modifying the AND circuit to keep advancing after 59, but only when the hour=04 and minute=20. This way, the clock works as normal for all times except 4:20. And when it's actually 4:20, the seconds will advance through 59 and get to 60. And 61, 62, and so on.

But we must make sure to stop it after 69, so we need another AND circuit to detect when the counter reaches 69. And more importantly, we can't just zero out the counter; we must force the next counter value to be 10, because otherwise the time is wrong.

It's very easy to zero out a counter, but it takes a bit of extra circuitry to load a specific value into the counter. But it can be done. And if we do that, we finally have counters suitable for 69th second operation. Because numbers 64 and higher require 7 bits to represent in binary, we can provide the 7th bit to the 7seg driver, and it will show the numbers correctly on the 7seg display without any further changes.

TL;DR: it can absolutely be done, with only some small amount of EE work