Tocqueville originally didn't talk of a mere "experiment" at all. Instead, the french original (De la démocratie en Amérique, Paris: Gosselin, 1835, p. 41) has the following lines (emphasis mine):
C'est là que les hommes civilisés devaient essayer de bâtir la société
sur des fondements nouveaux, et qu'appliquant pour la première fois
des théories jusqu'alors inconnues ou réputées inapplicables, ils
allaient donner au monde un spectacle auquel l'histoire du passé ne
l'avait pas préparé.
In fact, Henry Reeve translated this in the first english translation 1835 (Democracy in America, London: Saunders and Otley, 1835, p. 15) as follows, adapted also by John Spencer, and reprinted multiple times:
In that land the great experiment was to be made, by civilized man,
of the attempt to construct society upon a new basis; and it was
there, for the first time, that theories hitherto unknown, or deemed
impracticable, were to exhibit a spectacle for which the world had not
been prepared by the history of the past.
So, "great experiment" wasn't used by Tocqueville at all - he simply wrote "devaient essayer" - "should attempt (to build...)". Modern translations usually stick to to french original. So, A. Goldhammer translated in 1994:
Here civilized men would attempt to build society on new foundations ...
This correspondents to early translations in other languages, e.g. the german translation by F.A. Rüder (1836):
Dort sollten civilisierte Menschen versuchen, die Gesellschaft auf
neuen Grundlagen einzurichten...
So, in short: every edition that renders the famous Reeve / Spencer translation will usually contain the "great experiment" version. But i assume that a modern, critical edition would refrain from repeating that phrase, as its foundation in the french original is too weak.
It seems that Tocqueville itself was doubtful about Reeves's translation of "De La Démocratie en Amérique". In a speech about his translation, Goldhammer cites from a letter that Tocqueville wrote to Reeve in 1839:
Without wishing to do so and by following the instinct of your
opinions, you have quite vividly colored what was contrary to
Democracy and almost erased what could do harm to Aristocracy.
Additional note: there's a great overview of Alexis de Tocqueville's writings available online, provided by the UPenn library: http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Tocqueville%2C%20Alexis%20de%2C%201805-1859. There are also translations to different languages as well as the french original.