I am a research assistant in math in an institute somewhere in Eastern Europe. In contrast to western countries, you may have here a permanent (i.e. quasi-lifetime, ultra-secure) position even as you are still a PhD student. However, at that point you have a salary of ~$3.5k/year after taxes, which can be supplemented by being in a research grant or some other similar funding.
A productive day:
[06.30] My alarm rings, I wake up, morning daily routine.
[07.30] I leave home, I walk to the institute (somehow, home and work are positioned in a way such that any public transport lengthens the time for commute).
[08.20] I arrive at my office in which I am alone, turn on my computer.
[08.20 - 09.00] Read my mail, answer my mails, put in my schedule something I remember I have to do and could easily forget, check out RSS reader.
[09.00] I realize I don't have anything "urgent to do" (e.g. write an abstract for a conference, preparing a lecture) and may therefore work on an interesting problem.
[09.00-13.00] Productive work. This usually implies a lot of Googling and skimming through many, many papers to get a feel for the state of the art in a sub-sub-field of math. I may write a lot of summaries of theorems with pen and paper to understand what exactly are the differences between them and where are the main difficulties. I sometimes suddenly realize that there is something everyone missed and I might discover a new link. More frequently, I realize that my ideas are blind alleys. I may get some new ideas for that, then I search again for a lot of papers that may help me (becoming frustrated when some of the papers I want to read aren't even behind a paywall, but completely forgotten except for citations). When I'm very very well-read and very very very lucky, I find new natural definitions and interesting theorems and write papers about them.
[13.00-13.30] I go out, have some lunch at a canteen-like place nearby, get back to my office.
[13.30-17.00] Productive work like before. If everything goes well, I don't see anybody for the whole day.
[17.00-18.00] The road back home.
[18.00-22.00] I eat, I read, I watch movies et al.
[22.30] Shower and then sleep.
An unproductive day:
[07.30] My alarm rings later because I am very tired from something I did the last day so I had foreseen I might sleep more. I am somehow not yet rested so I go back to sleep.
[11.00] I finally wake up and I realize I have to go to the institute because I have some form to fill.
[13.00] I arrive at the institute, I fill the form, I eat something like in the above and then go to the university because I have some tutorial to teach.
[14.00-16.00] I teach.
[16.00] I realize I can't do anything interesting that day so I head home and feel a bit bad about myself.
[16.30-end] Like in the above.
Most of my days are a mix between the two "ideals" :)