PCB Design: Through hole components on both sides?

If you plan on hand soldering, through hole parts on both sides is doable.

The issue with manufacturing though is it is difficult to flow solder with through hole parts on both sides. You can do it, but you may need to do a lot of conformal solder masking, once for each side, and that is labor intensive and expensive. Some fab houses have specialized equipment that allow more selective soldering, but there are setup costs involved in that, so unless it's a big run, the cost per board factor is significant.

As with all double side populated boards though, the space saving is limited by the routability. On a dense board, using the other side does not buy you as much space as you would imagine and adds considerable cost.

Further, since through hole parts are already effectively double sided, the leads poke through to the other side, you cant re-use the spots where things poke through, and you need to be able to see those leads to solder them. So again, it saves you very little.

Using SMT instead of through hole is a better way to reduce the size.

If the board is still too big with both side populated SMT, your next best bet is to split the board into two with suitable connectors so you can make it a sandwich. That can be designed with both parts on a single panel and manufactured as a single board and split and assembled later. Another alternative is to build it on a flexible circuit and fold it up.

This won't gain much density...

In general, double-sided load for density's sake is only worthwhile when SMT comes into play -- either as a mixed-technology board as is commonly seen in consumer products using wave soldering, or as a mostly SMT double-sided load as is found in higher-density boards with BGAs and such where a reflow/selective combination process is used. As Trevor points out, most of the space is taken up by the pad areas, which can't be overlapped anyway. Furthermore, trying to do a double sided load with parts opposite one another raises issues of part-to-board clearance vs lead trim, never mind the severe difficulties with sequencing stuffing and soldering steps, which can even force a board to be hand assembled or partially stuffed, soldered, then stuffed more and soldered again. Both of these are killers in production.

But it can be a gain in layout simplicity

I have, however, done a double-sided load in a fully THT design. Why? Because putting parts on the back side of a board can be a big help with getting busses the right way around. Having to go from IO0 to D7 and then Q7 back to DQ0 can make for a confusing schematic; atop that, backannotation of this sort of thing is something that not all tooling supports particularly well. In a hand-assembly situation, it's easier to slap the offending part on the bottom of the board, especially if your layout tools have poor backannotation support, as I mentioned earlier.

One can imagine putting one or two extra-large THT components on the bottom side and the rest of the electronic on the top side. To achieve that you will need to order a solder pallet (an adapter) for your PCB to hold the parts in place during soldering. Extra cost between $/€ 700 and 2000 + some additional production costs. But the THT components must be realy big and the advantage obvious to make it worth it. A solder pallet or another sort of self-made adapter may be needed also for manual soldering.