All grain - is full wort boil absolutely necessary?

Good answers already given, although if you choose to boil a higher gravity then add top up water, keep in mind that that boiling a higher gravity wort reduces hop utilization, so you need to use more hops to achieve the same level of bitterness. (Doesn't apply to flavor and aroma hops.)

Typically it's between 10% and 30% more. The table here lists hop utilization factors by time and gravity. If you're not using brewing software, which can do this calculation for you, you can work out how much more hops to use with the formula:

Utilization at Original Gravity
------------------------------- - 1 * 100 %
  Utilization at High Gravity

For example, if your original recipe has 60min hops for a 1.050 wort, but you do a concentrated boil, with a 1.080 wort, then the amount of additional hops is:

 = (0.231 / 0.176 - 1) * 100 
 = (1.3125 - 1) * 100
 = 0.3125 * 100
 = 31.25 %

So, you'd need about 30% more hops than in the original recipe if you boil 1.080 wort rather than 1.050.

A full wort boil is not absolutely necessary, but you shouldn't be topping off too much of your volume. Your efficiency will suffer greatly from topping off. Even if you're able to hit 80% effiency with 11.5L of wort (a big if), after boiling and topping off to 23L you'll end up with below 40% brewhouse effiency. You would also definitely be limited in terms of how many pounds of grains you could work with.

This isn't an issue with extract brewing, because you get the same amount of sugars from the extract no matter how much water you use. This is not the case with grains at all, since you'll need more mash water, more sparge water, and you definitely need to boil all the wort you obtain.

Some ideas to try out all grain with your current setup:

• Try partial mash, adding only a couple of pounds of extract. You can learn the ins and outs of all grain brewing without much of the hassle. You'll get to mash grains and save some money.The best guide I've seen for this is DeathBrewer's, and my homebrewing process is based on his.
• Try a smaller batch size with a nearly full boil. Scaling recipes is incredibly easy. For example, you can take a recipe for 23L and make it into a recipe for 11.5L by simply cutting all the ingredient amounts in half. I scaled back my batch sizes to 4 gallons for reasons similar to yours.
• Try splitting your batch into two simultaneous boils. If you're set on a 23L batch, you can go through your boil in two pots, adding equal amounts of hops and combining them later.

No, you don't have to boil the full volume in AG brewing. I only had a 7 gal. pot when I started AG so I's boil about 5 gal. down to 3.5-4, then add top off water. Boiling less will reduce your efficieny because you don't collect as much wort. You need to use more grain to make a higher gravity wort at less volume, so you can top off afterward. That was my main motivation to get a bigger kettle and do full volume boils. But I won some awards with partial boils, so if that's all you can do you can still make great beer.