Which of the following has maximum nucleophilicity?
According to me, the answer should be (C) in general cases but this might get changed to A) in special cases. The best way to judge their nucleophilicity order  is to check their basicity. The weak bases are generally better nucleophiles in polar protic solvents . The acidity order of the  conjugate acids are $$\ce{CH_3OH &lt; (CH_3)_3COH  &lt; NH_3 &lt; CH_4}$$So, the strongest acid is $\ce{CH_3OH}$ and weakest acid is $\ce{CH_4}$. So, weakest base is $\ce{CH_3O^-}$  and strongest is $\ce{CH_3^-}$. So, the maximum nucleophilic character should be of $\ce{CH_3O^-}$,not generally $\ce{CH_3^-}$ as basicity is inversely proportional to nucleophilicity in general cases. Coming to D), the inductive effect of three methyl groups actually decreases the acidity of tert-butyl alcohol. So, the tert-butoxide ion is more prone to take $\ce{H^+}$ rather than to nucleophilically attack.  It also creates steric crowding, and preferably takes easily ionisable hydrogen ion if possible from the compound.So,  looking only at increased $\ce{+I}$ effect doesn't tell that its nucleophilicity is increased. About A), this $\ce{CH_3^-}$ is usually seen in Grignard reagents, whose basic activity is much much greater than its nucleophilic activity. If acidic hydrogen is present in the compound, it preferably takes that. But, if a compound doesn't possess any acidic hydrogen (special case), then its nucleophilic character is more as being less electronegative than $\ce{O}$, it is very less reluctant to keep yhe lone pair on itself like the case in $\ce{CH_3O^- or (CH_3)_3CO^-}$. Thus in the absence of any acidic hydrogen $\ce{CH_3^-}$ is better nucleophile. So, from the above discussion, it can be concluded that in many of the cases, where the attacked compound doesn't possess any acidic hydrogen, A) is maximum nucleophillic. But in the opposite  cases, C) is maximum nucleophilic.