Is it really that likely that something will go wrong? I can understand after 5 years but 3 years seems short.
It is unlikely that something will go wrong. You should see this more as an insurance (for technical defects only) and quality of life thing (better service). Would you rather spend some extra money now and be sure to have a working machine for 3 years, or would you rather save the money, keeping in mind that if a repair should be necessary, it can easily cost 2x to 4x the cost of AppleCare?
So far I have gone without. Including my wifes machines, I'm on MacBook #5 now, none of them with AppleCare:
- 2007/08 non-unibody MBP: Died at 38 months (outside AppleCare), repaired for free due to 8600M GT graphics card issue. No problems before or after that
- 2009 white Macbook: My dad is using this now. No problems ever
- 2011 15'' MBP: Some issues with the screen that were there from day 1 (i.e. would have been covered under the 1 year warranty), still running fine after years of heavy use
- 2012 Air: No problems
- rMBP bought in 2014: no problems
Combined we would have spent probably $1k - $1.5k on AppleCare. That is a full new laptop. In particular on the cheaper models I feel it is not worth it. However I would consider it for machines north of $2.5k and in particular if they come with discrete GPUs. They seem more likely to fail in the long run.
PS: Note that when the 2007 17'' MBP failed I was offered a flat-rate repair for $330 or so. It made what would have been a $700+ repair reasonable. Only later it turned out that the GPU was the issue and the fee was waived.
If Apple still does this then it provides a small safety net.