More details about the output connections available on your amp and pedals would help, but in the meantime I have a couple suggestions:
Most multi-effects pedals have a headphone out, or even a line out, both of which could be connected directly to your Mac Mini (from looking at the specs, it looks like your Mac has a 3.5mm stereo line input). In this case you would likely need a stereo 1/4" to stereo 3.5mm cable to connect the pedal output directly to the computer (usually headphone/line outs on pedals are 1/4", but some are 3.5mm, be sure to confirm on yours). This should produce decent results for simple recording and getting some ideas down.
Next option is using the amp's line out, or direct out, which not all amps have. If your amp does have one, it is probably mono but you can still connect it to your Mac line in with a similar cable. The benefit of using the pedal instead of the amp is that you will likely get a stereo signal from a pedal, which can add some dimension to your recorded sound depending on how good your pedal is.
While you mentioned you don't want to spend a lot of money an interface, there are a couple options you still have. Behringer makes a USB guitar interface which would take your guitar signal straight to your computer. They also have a standard USB interface which would be better if you want to use signal from the amp or pedal. (for that one you would likely need some sort of 1/4" to RCA cable to connect amp/pedal to that particular interface).
Finally as I mentioned in the comments, there is the IK Multimedia iRig, which you plug your guitar into and connects it to your iPhone/Pad. I've heard great things about this particular option, especially because it comes with effects and amp models built in, so you don't have to worry about connecting your pedals or amp but you can still get a wide variety of decent tones.
Again, this all depends on what kind of outputs your amp and your pedal support, so more info on what gear you are using will help narrow this down.
Edit: since you've got a GT-6, that would probably be the simplest way to connect to the computer because it has built in amp and speaker modeling. You want to use the pedal's left/right output going to the Mac's input. For this you should use a dual 1/4" mono > 3.5mm stereo cable to get both left and right channels of your pedal to your computer. You're going to have to mess around with the output level knob next to the output jacks to get an ideal signal. Also you might be able to use the digital out because the Mac Mini also accepts digital audio input. Not 100% sure on compatibility though.
Here is some great info on the subject:
Recording Electric Guitar with PC / GT-6
Recording guitar through Boss GT-6 into Soundcard
Your best sounding option is going to be to mic the amp through an audio interface. But if you don't want to invest in that setup you are going to have to carefully edit the patches on the GT-6 to make sure they sound good on their own (use headphones). There is one more option, you can record the guitar dry (through pedal to Mac), and use free VST guitar amps/effects to get a great sound.