I'd not like to ruin your learning fun BUT if you get a more or complete idea on this project you can move on to more difficult ones.
The circuit below is almost exactly what immediately came to mind for me (I have had lots to do with 4017's in recent years :-) ) and lo and behold somebody has done a very nice job of writing it up.
The 4017 is a decoded "Johnson Counter" (look it up) which provides a sequencing one-of-ten output.
You can cause it to
count up to position N and stop,
or to position N and then reset
or you can chain chain several together
or more ....
A very useful IC.
Datasheet for the basic CMOS version here
and for the buffered 74HC4017 version here.
Note that the "basic" CD4017 has a very special feature which tends to be lacking on all "improved" versions - it has a Schmitt triggered clock input - which means that you can use it with a user pushbutton input or other slow and noisy input. ometimes an immensely valuable feature.
The circuit itself is enough:
Does this do whta you want?. Well, almost.
Look at the enable and reset lines.
Look at the datasheet.
What happens if you plug the enable line into output N?
https://homepages.westminster.org.uk/electronics/images/4017_08.gif
BUT they have done a really superb job of presenting a plug in bread-board version here
Leading to this. You could use one small breadboard and less LEDs and a different oscillator
(eg 555 / 4040 et al etc) but this is an extremely nicely done example
THEN you can consider a zillion alternatives [fromhere] - all images hotlink to a page. Look at he top of the page to see the obvious and extremely useful way that I got this eggtimer circuit collection and this overlapping but not identical egg timer circuit collection (plus some other stuff in each case).
74HC4017 "under the hood":
Clock accuracy:
Try it and see.
Use a good quality clock cap- NOT a ceramic.
What if you clocked it twice as fasts and used eg diodes to OR a single LED per 2 outputs?
Or 3 times as fast?
If you want to use a faster clock look at CD4040, CD4020, CD4060. Note that one of these can both divide and self oscillate. You can still have 2 ICs total but a clock and a divider as well. Enjoy.
I agree with Olin: ditch the 555 and get a microcontroller. The PIC10F200 is the first one which comes to mind if you only need a couple of I/Os and a simple program. Thanks to the internal oscillator it doesn't need any of the external components required by the LM555. The 10F200 is only a few cents more than the LM555 and can easily replace two of them. The LM556 is no competition either: it costs twice what you pay for the 10F200.
So Olin's answer is the right one? No :-). I'm going for the bonus! I recently found the Atmel ATTiny5, which is comparable with the PIC10F200: same SOT23 package, same 512 bytes program memory, internal oscillator. BUT! The ATTiny5 has an ADC, which the PIC doesn't have! Connect the potentiometer to it and the bonus is mine! :-)
(You can also make a crude ADC with the PIC, but it needs 2 pins which you can use as outputs, and since we already have two outputs and one I/O of the 10F200 is only input, you'll have to give up that buzzer output. Edit. On second thought, you can do it with one I/O, but you'll need two resistors an a capacitor instead of one resistor and a capacitor.)
The software will be very much the same as for the PIC: program an LFSR (Linear Feedback Shift Register):
An 8-bit LFSR can cycle through up to 255 combinations before it repeats, a 16-bit LFSR through 65535. Use the potentiometer reading to define the clipping values for the timer.
Best Answer
Here's one that's all digital, needs no tuning, and runs on any 1 second clock oscillator.
HOW IT WORKS:
C1R1 is a differentiator, and as V2 first comes up, a narrow positive-going spike (MR) will be generated across R1. It's used to make sure that when power comes up, U1,U2,U5 and the R-S latch comprising U6A and U6B are all in known states, with all the counter outputs reset to zero and the latch set, which will turn K1 ON. Then, when the next clock comes along, the 12 minute counter (U5) and the 48 hour counter (U1 and U2) will both start counting up, simultaneouslhy with V1 being the 1Hz clock source.
When the 12 minute counter gets to 720 (the number of seconds in 12 minutes) U1A,B,and C will decode that state and send a pulse to U5B which will RESET the latch and turn the relay OFF. At the same time, U6A's output will send a high to U4B which will force the counter into reset and hold it there until the 48 hour counter counts to 172800, the number of seconds in 48 hours.
When the counter gets there, U3A,B,C, and D will decode that state and send a high to the counter's RESET pins, forcing all of its outputs low and starting a new 48 hour counting cycle. The pulse is also sent to the latch, which it SETS, turning the relay ON and releasing the RESET on the 12 minute counter, starting the new 12 minute cycle anew and in sync with the 48 hour counter.
So, in brief, the relay turns ON and both counters start counting on power-up. 12 minutes later, the relay will turn off and stay off until the 48 hour counter times out, when a new cycle will start, seamlessly, with the relay turning ON and both counters starting their countdowns, all simultaneously.
ON THE DECODERS:
The 12 minute decoder:
Since 12 minutes is 720 seconds and U5 is a binary up-counter, once its outputs have been cleared and it's allowed to count, when it accumulates 720 one-second clock pulses its outputs will look like:
$$\style{color:black;font-size:100%}{0010\ \ 1101 \ \ 0000}$$
With the MSB leftmost.
In order to detect/decode that unique state, and use it to our advantage, all we need to do is to AND all of the counter outputs which are ONES when the count reaches 720, and use the output of that decoder to get done what needs to be done before the next clock comes along. Not a big deal with a 1 second clock.
The 48 hour decoder:
The logic for the 48 hour clock is similar, but when it counts up to 172800 seconds, its outputs will look like this:
$$\style{color:black;font-size:100%}{0000\ \ 0010\ \ 1010\ \ 0011\ \ 0000\ \ 0000}$$
So the The 48 hour decoder's output will then go true when the five output ONEs are ANDed and used as a trigger.
If you're interested in the circuit, Here are the files you'll need to run a simulation, using LTspice, if you're so inclined...
If you are, just download all of the files into the same folder and left click on either of the .asc files. If you've got LTspice installed on your machine it should find the file and bring up the schematic. If you don't, it's available, free, at http://www.linear.com/designtools/software/
As an aside, the "test" schematic is identical to the main one, with the exception that the decoders have been removed so that a few cycles can be run to check the logic without having to wait forever for a solution.
Enjoy!