To try to update the list as we go... and to presume we're talking about Test matches only as domestic cricket could see many better ones...
(I've also added in all ground names and cities if not named after the city, and countries in West Indies, plus full names, exact figures, opposition if for a home player, and season. I've also added Edgbaston and Basin Reserve)
MCG, Melbourne - for me I'd go for Bob Cowper, 307, vs Eng, 1965/66
SCG, Sydney - Brian Lara, 277, 1992/93
Adelaide Oval - hard to look past Don Bradman, 299*, vs SA, 1931/32
The 'Gabba, Brisbane - Richard Hadlee, 15-123, 1985/86
The WACA, Perth - EITHER Ian Chappell, 156 OR Roy Fredericks, 169, Aus vs WI, 1975/76
Bellerive Oval, Hobart - just has to be Kumar Sangakkara, 192, 2007/08 (and this counts as a * on first-chance too, as it was only ended by an appalling decision)
The (Kennington) Oval, London - tough one between Len Hutton, 364, vs Aus, 1938 and Muttiah Muralitharan, 16-220, 1998
Edgbaston, Birmingham - I'd go for Graeme Smith, 277 + 85, 2003
Old Trafford, Manchester - Jim Laker, 19-90, vs Aus, 1956
Lord’s, London - cannot be anything other than Don Bradman, 254, 1930 (just a reminder - every shot in this innings went exactly where he aimed it); though this is tough on Bob Massie, 16-137, 1972
Trent Bridge, Nottingham - just has to be Alec Bedser, 14-99, vs Aus, 1953
Headingley, Leeds - Graham Gooch, 154*, vs WI, 1991; OR Ian Botham, 149* & 6-49, 1981
St Georges Park, Port Elizabeth - I'd go for either Andrew Strauss, 126 + 94*, 2004/05 or Mathew Sinclair, 150, 2000/01. On balance I'd lean toward the latter, though I watched close to every ball of the former and the assurance was staggering.
Newlands, Cape Town - I'd have to go here for Herschelle Gibbs, 228 (off 240 balls), vs Pak, 2002/03, though I'd also mention Johnny Wardle, 12-89, 1956/57
Kingsmead, Durban - I'd place my vote on Gary Kirsten, 275, vs Eng, 1999/2000
The Wanderers, Johannesburg - don't agree at all with the Gilchrist 204* in 2001/02 myself, I'd say Michael Atherton, 185*, 1995/96 was a far better innings. Not only was it chanceless (no, Kirsten was NEVER going to catch that one at short-leg on 99, whereas the one to Kallis at slip on 20-odd was straightforward) but it saved the game against a fresh and highly potent attack rather than battered a woebegone one which had already suffered from a flat deck and their fielders' ineptitude.
Centurion Park - in this ground's brief history I'd say the best (even though it was a dead game) came from Allan Donald, 8-96, vs Aus, 1996/97
Eden Park, Auckland - the Hammond knock was dubious in many ways to me, so personally I'd go for Daneil Vettori, 12-149, vs Aus, 1999/2000
Jade Stadium, Christchurch - Nathan Astle, 222, vs Eng, 2001/02
Basin Reserve, Wellington - absolutely no question, Courtney Walsh, 13-55, 1995/96. Weak NZ batting this undoubtedly was, but I can't believe there've been many cheaper 10-fors this side of 1900.
Kensington Oval, Bridgetown - Brian Lara, 153, vs Aus, 1999
Queen's Park Oval, Port-of-Spain - Allan Border, 98* + 100*, 1984; well, maybe. However I'd also give a mention at least to Hanif Mohammed, 337, 1958 and Lawrence Rowe, 302, vs Eng, 1974
Bourda, Georgetown - I'd give this to Imran Khan, 11-121, 1988
Sabina Park, Kingston - this obviously has to be Alec Stewart, 9*, 1998. Seriously, though, that was easily the best single-figure innings in Test history, and those who remember the game will know why. But the best surely has to be Garfield Sobers, 365*, 1958
The ARG, St.John's - has to be Wasim Akram, 11-110, 2000
Eden Gardens, Kolkata - VVS Laxman, 281, vs Aus, 2000/01
The Chepauk Stadium, Chennai - Virender Sehwag, 319, 2007/08, vs SA
The FSK, Delhi - Anil Kumble, 10-
Hyderabad - Sachin Tendulkar, 186, vs NZ, 1999/2000 ODI
Mumbai (Wankhede or Brabourne) -
The Chinnaswamy Stadium, Bangalore -
The Gaddafi Stadium, Lahore -
Peshawar Cricket Ground -
National Stadium, Karachi -
Rawalpindi Cricket Stadium -
Iqbal Stadium, Faisalabad -
Colombo (SSC) -
Colombo (PSS) -
Colombo (RPS) -
Colombo (CCC) -
Galle International Stadium -
Asgiriya Stadium, Kandy -
Harare Cricket Ground -
Queen's Sports Club, Bulawayo -
Dhaka
Sharjah Cricket Stadium -
Will let some other people try their hand at the remaining.