What specs would you recommend for an OSM tile server dedicated to serving tiles for a single, small country (e.g., Portugal only)? Total server load is estimated to be no more than two/three users accessing tiles at the same time.
As mapperz pointed out, your load is probably small enough that anything you have will do. If you have some flexibility though, try for fast disks (ideally SSD) and more memory. I suggest fast disks and memory because these are likely to be the limiting factors (for the postgres database instance).
It is interesting to look how openstreetmap does it: http://wiki.openstreetmap.org/wiki/Servers documents their servers. Given their load, it isn't really a lot of machines doing the tiles work...
For users who are looking for such requirement. Here is simple solution to create all openstreetmap tiles for a single country osm-for-my-country. There are scripts for all tasks on a Linux machine:
Install all necessary software like osm2pgsql, postgres, mapnik, kosmtik etc
Download your country data
Download and install map style openstreetmap-carto
Create all tiles for your country
Update tiles on daily basis
Serve tiles with node.js or you can serve these offline map tiles with any simple http serevr or apache