Interesting question, but I doubt that any club that has been well capitalized over the years has completely resisted the urge to tinker with their putting surfaces. In my rather limited experience, most of the clubs with a very original look to their greens could tie their restraint to one of two factors: First, a lack of cash. Or, second, the presence of another course or two that got all of the attention.
I'll cite three from Chicago. Olympia Fields at one time had four golf courses. Currently it has two, the North and the South. The North was the fourth course built and it hosted the US Open in 1928, I believe. Over the course of time, its greens were altered on several occasions, usually for trying out a new seed. A few of the greens have been rebuilt as well. The South Course consists of 15 holes from the original #1 course, two from the other courses that were eliminated and one entirely new hole. I am pretty sure that as of the time of the recent renovation three years ago, those original greens were untouched, save for expansion that was meant to bring the greens out to their original size and shape. The North Course got all the attention, while the South just stayed the way it always was, with some limited exceptions. As the North Course struggled with the decisions to regrass the greens, the South greens were always very consistent. In retrospect, some of the well-intended mischief on the North did it no favors while benign neglect served the other course quite well.
Another example is Medinah, which has three courses. The #3 course has been redone more often than Joan Rivers' face and there's just no way to recognize the original greens. Which is just fine, actually, because that course seems to exist for two reasons: To host professional tournaments and to be the money maker for the club's revenues from greens fees. The Number 1 course is probably going to get renovated after the 2012 Ryder Cup. A handful of the original greens were completely altered in a way that is inconsistent with the original look of the place and it will be fun to see a consistent design there. The interesting observation about Medinah, a place with nothing but money over the years, is that the #2 Course has been largely ignored in terms of updating, as the membership kept fiddling with the championship course and occasionally tweaking the main members' course, #1. the #2 course has been known as the "Women's Course", but it is a charming and eminently playable and tight routing with terrific old time Tom Bendelow greens. Benign neglect also seems to have helped here.
Then, there's an old course in south suburban crete, Lincolnshire, that is a Tom Bendelow design. The club has probably never been flush with money and their greens, to my recollection, are relatively unscathed, with a bunch of small, frisbee-shaped surfaces with severe back to front and occasional side to side pitch. A fun place to play old time golf.
Shoreacres may be representative of another form of benign neglect, in a way. It has always been an old-fashioned, quiet kind of place with a decidedly non-meddlesome membership regarding messing with the golf course. Tom Doak has done some minimalist retooling of the place, so it would be interesting to learn if those greens had been significantly altered in prior decades.