Ceviche, you did say "Budget is not an issue." Thems is some pretty strong words
But seriously, there are various themes related to a house, and for me, mostly positive. The biggest reason I'm still in Recoleta and not in a house in Colegiales, or Belgrano or something is that everyone in my family either works or goes to school somewhere within 5 blocks of 9 de Julio.
I find apartments to be horrible for the most part. Too much noise going on above, below, to the side, etc. At least in a place where you can usually find a house you only have to deal with traffic and party noise, although a bit more laid back in many "residential" neighborhoods. Even houses that are attached to others (common in the city), you have only two sides, not four, to worry about noisy neighbors.
I used to live in a 250 sq mtr house in a closed neighborhood, outside the city. I had a pool and about 1200 sq mtrs of property to go along with it. I employed a pool guy to keep the pool clean, and a lawn guy to take care of the plants and grass and such. Total cost me about $100-150 USD a month. We had enough people in the house to take care of cleaning and such, and either I, or someone I hired, took care of little maintenance issues. Not a big deal at all.
Ceviche, there are houses in the city that are very doable and actually have not so much grass to worry about
I have a friend who owns a 500 sq mtr house near Paraguay and Uruguay, not 5 blocks from where I live. It has a little patio out back on the ground floor and a roof (which he's thinking about putting decking on top of) with a small efficiency apartment on it. Wide lot, 3 floors + roof, 5 bedrooms, 3 bathrooms in-suite, huge den, huge dinning room, etc. I think that's probably not what you'd be looking for, but that was just an example of houses that are not in the sticks without a lot of exterior maintenance - although interior maintenance would be something else for a house that size.
I've looked at places in Colegiales and Belgrano that were nice little houses, from 100 to 150 sq mtrs, in quiet neighborhoods. I know someone who found a 300 sq mtr house with a pool and a quincho in back and a little bit of grass, for about $1500 USD a month.
You might also be interested in a PH (I still don't know what the letters stand for, after 8 years here!). Those are like narrow houses, usually two floors, going from the front to the middle of a block or something (various configurations). They are all over the place. I had a friend who had one in Colegiales and it was quite nice, including a sun deck on top with a parilla. We used to have some pretty cool asados over there.
Don't let the house nay-sayers influence you! If I could find a cheap house in Recoleta or thereabouts I'd jump on it!
But I'd never go live in a closed neighborhood again, I don't believe. Too enclosed (walls, gates, armed guards checking you in and out in a car), apart from everything (though usually pretty safe) and I found the class of people who live and own homes in those places to be a bit too snooty for my tastes - and they don't like Paraguayans!