How do I create a very long delay with Redstone?

The longest I can think out of top of my head?

Fill the dropper with as many items as many days you want the signal to take. Near each noon the dropper will discard one item, for a max of 576 in-game days. If that's not enough, you can restock the dropper with a chain of hoppers and a double chests, and only when all the items are exhausted the signal will be produced.

If you want this retriggerable, with limit of 320 days you can "discard" the items into a hopper blocked with a piece of redstone that will be deactivated once the time is past, and feed all the items back in.

If you want something faster, clock that dropper with a pair of repeaters or any of hundreds minor clocks. With an adjustable clock and a specific number of items you will easily fit within that 3-5 minutes range.

Use several clocks and an AND gate

A clock that takes 7 ticks and one that takes 8 ticks will output every 56 ticks, using only 5 repeaters instead of 15. Using this method you can use 6 repeaters to get a 42s (420 tick) delay.

You simply have a clock that takes 3 ticks (1), a clock that takes 4 ticks (+1=2), A clock that takes 5 (+2=4), and a clock that takes 7 (+2=6 repeaters).

3 x 4 x 5 x 7 = 420.

Scott Chamberlain's answer is pretty good but if you don't like the complexity of pistons, here is a compact T flip flop without them:

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Here is a ~2 minute long clock:

Note that each extra T flip flop will halve the clock speed. Also note that a small change in the initial clock's repeaters will greatly affect the result.

I would recommend using a slow automatic clock (See B') tied in to a chain of T flip flops, (if you have access to sticky pistons you can make them extremely compact, see the Z layouts).

In the above diagram A would take 2x long to cycle, B would take 4x long, c would take 8x long, D would take 16x long.

As you see it gets very long very fast, I doubt you would need more than 4 or 5 T flip flops to get a 2-5 min delay.

for 5 minutes exactly you could have a dispenser drop an item onto a pressure plate, and the item would disappear after 5 minutes

I also have a video of what I use when I need long pulses on my youtube channel

You could modify this to use a system to activate the circuit several times sequentially to save space. I may do a video on that again some time.

I've recently worked out this circuit:

On the right is one of Etho's Hopper timer, which can be extend to about three and a half minutes, with precise adjustment by adjusting the number of items in it. On the left is another hopper timer, just like it. The timer on the left is not being used as a timer, but rather, a counter, counting the number of times the timer on the right has cycled. The redstone on the hoppers on the left disable the counter except when the timer on the right resets, and then allows it to run just long enough to allow one item to shift around.

When all of the items have shifted in the counter from one hopper to the other, it switches the output on the far left 'on', and all the items have to shift back to the first. After a full round trip, the clock and the output turns 'off'

Since the counter has a capacity of 5 stacks of up to 64 items, and the input clock has a maximum delay of 3.7 minutes; this clock can be adjusted to have a cycle time as long as 40 HOURS!. If that's not long enough, add another counter to the output of the first; which will top out at a delay of almost 3 years.

Minecart tracks can easily be extended to provide a very long delay, taking up (relatively) little space. The space they take up can easily be put anywhere you want too, which will avoid clogging up your circuit.

Wire up the output of this circuit to a flip flop, and have your input connected to a cart dispenser. When you hit toggle, a cart is released. 5 minutes later it will arrive and switch the flip flop. You can then return the carts to the dispenser.

Add more tracks as required

I'd recommend tight spirals of track to extend the time, which ensures they won't stray into a chunk that might get unloaded.

A 2-hopper clock. Immediately beside it a redstone t-flip flop, that uses the redstone to toggle the active hopper. a single falling-edge pulse emitter using a comparator-repeater feed off of one side of the hopper clock. that emitter feeds back around to the input of the flip flop. a single feed from your daylight sensor between the pulse emitter and the flip flop. This is how it works. One side is fed with as many items as it takes to count out half (HALF) of the time you need. The clock is normally at rest. An 'on' signal from your detector flips the t-flipflop, activating the hopper clock. items xfer from one side to the other. once full, a signal from the comparator-repeater off your hopper clock activates the falling-edge pulse emitter, toggling the tflipflop, and thus your clock. the other hopper empties back. The clock works for one full cycle, then halts, waiting for another input from your detector and is FULLY customizable on the timeframe. I use this setup to make a bridge across lava that only stays up so long after a button press.