It always struck me that Scotland was a bit under-represented in the greatness stakes, especially for a country with quite a few relatively old universities, so don't have the excuse of lack of education system.
too many great ones to mention and as for what they invented how about.....
adhesive postage stamps anaesthetics antisepsis artificial diamonds reaping machine
Bank of England latent heat Brownian movement Buicks chemical bonds
penicillin the decimal point documentary films Encyclopedia Britannica engineering sciences
fax machines first cloned mammal flailing machines geosciences golf
historical novels hypodermic syringes Kelvin scale percussion powder logarithms
Maxwell's equations marmalade mackintosh raincoats macadamized roads microwave ovens
colloid chemistry breech-loading rifle tubular steel quinine Sociology
pneumatic tyres pink bathtubs hollow pipe drainage Peter Pan radar
paleobiology polarization cure for scurvy
refrigerators Neptune bakelite iron bridges solitons
the steam engine telephones thermos flasks/dewars the telegraph television
the stereotype sulphuric acid the steam-hammer cure for insomnia paraffin
Whisky US Navy Chilean Navy Economics Cloud Chamber
When you start looking closer at these 'inventions', much of it turns out to be propaganda.
For example, Scots are taught at school that one of the greatest Scottish inventors is John Logie Baird, inventor of television. In fact, he is one of several early pioneers of television from a number of countries. The Americans claim an American invented it, the Russians claim a Russian invented it. That's because the guy usually credited (outside Scottish schools) with inventing television is Zworkin, a Russian who moved to America. See eg. here: http://www.physlink.com/Education/AskExperts/ae408.cfm
Most countries seem to be able to claim that they invented most things, with a little creative accounting. Often scientists in more than one country come to similar conclusions more or less simultaneously. Undoubtedly there were some decent Scottish inventors - Alexander Fleming is a notable example. But pretty much every civilised country has many excellent inventors.
I'd say Burns was the only really decent writer (OK, RLS wrote decent adventure stories). No particularly notable composers. Painters? Well, I rather fancied the enchanting eyes of John Singer Sargent's portrait of Lady Agnew of Lochnaw, but there's precious little of international standing.
Simon, I suspect and hope that you are being satirical. Scotland arguably has made a disproportionate contribution to society for a country of its size, but your lack of awareness of them does not diminish these achievements. Nor does Scottish success diminish achievements from other countries.
In one of the areas that you mention, "inventors" (science, engineering, medicine etc), there is little doubt that Scots have made a grossly disproportionate contribution. Unparalleled.
Writers, deid and current, hold their own across a wide range of genres. To be honest, a list would be very, very extensive, somewhat boring, and, if your mind is already made up ("RLS wrote decent adventure stories" - please!), not terribly helpful.
"No particularly notable composers"? Well, outside of John Adams (US citizen), who IMHO will probably be recognised as the greatest composer of the period (post Stravinsky, Schostakovich, Britten, Tippett etc), people like James MacMillan and Peter Maxwell-Davies hold their own and are of world stature.
Painting? "Precious little of international standing"? Aside from Raeburn, the Naysmiths, and the painters of the earlier periods, the 20th century Scottish Colourists like Peploe, Cadell, Fergusson, Hunter etc are internationally recognised as a distinct "school" of major importance. Joan Eardley? Byrne and Bellany are both current major artists of the present period, as are the younger "new Glasgow Boys", Campbell, Wiszniewski, Howson etc. This present crop, whose style I'm personally not too keen on, are about the hottest stuff on the international art market. Unless you could bid against Jack Nicholson or Madonna, you wouldn't even get a look in. But then, it's probably fair to say that modern Scottish painting has remained rooted in a figurative tradition and has largely ignored a lot of the short-term fads - so you may not be aware of them.
And the achievements would have been greater if so many didn't have to go away and run things for other people.
(BTW - edited the above about John Adams as it may have unintentionally appeared that I was claiming him for Scotland. Although, mind you, that surname...........)