finally!... I bought a 60 euro, 4 inch android smartphone without a data plan on it, and I already tried to download a OSM export to just browse it on my computer, without any luck. I thought I could export tiles and browse them as a html file on my phone, but I did not manage to do it.
On google map on the nexus 7 I managed to select a portion of an area and download it, but I could do it again on my huawei Y330. I could not make it work offline.
Huh no you're thinking way too difficult, just download the OsmAnd app. I'm afraid it needs a few gigabytes of storage though, at least in Europe the map is very detailed.
They offer regions as default downloads, but it's all open source and open data, so if you want you can export a single city and use it in the app. It is quite a bit more work than just downloading regions (or small-ish countries like the Netherlands) though. See OsmAndMapCreator if you want to do this for the OsmAnd app.
There's a whole ecosystem of tools for dropping tiles on a device. This is a tool for creating them, but it has a handy list of devices where you can use them:
On google map on the nexus 7 I managed to select a portion of an area and download it, but I could do it again on my huawei Y330. I could not make it work offline.