A nucleophile acts by donating a pair of electrons to another atom's nucleus. In general, a negatively charged compound is going to be a stronger nucleophile than a neutral compound. In addition, as one proceeds down a given column of the periodic table, the nucleophilicity increases because the electrons are not held as tightly to the nucleus (electronegativity decreases).

is the best nucleophile, because it has a negative charge (more electron density), and its electrons are held less tightly than those of because sulfur is less electronegative than oxygen.

There are two major types of nitrogen-containing moieties in this molecule.

First, there are the aromatic nitrogenated groups, such as the purple, green, and gold. All three of these nitrogens, when reacted with an electrophile such as Boc anhydride, would produce positively charged species. This alone would be unfavorable, however, as these nitrogens each donate a lone pair to their aromatic systems, donating this lone pair to an electrophile would break the aromaticity of the system. Breaking aromaticity is always highly unfavorable, and hence, none of these three would readily react with Boc anhydride.

Second, there are the aliphatic nitrogenated groups, such as the red and blue. Of these two, the red is tertiary and the blue is secondary. This means the red would produce a positively charged, tetrasubstituted product when reacting with Boc anhydride, whereas the blue would not form a charged product. The blue amine is also more sterically available, and is the correct answer, as it has the best ability to act as a nucleophile. 

The molecules are almost exactly the same, except that each molecule contains a different group 6 atom. Size increases as we move down the group 6 column, and therefore nucleophelicity increases. Larger and less electronegative atoms hold onto their electrons more loosely, and are stronger nucleophiles.

The periodic trends of electronegativity and charge stability are useful tools for predicting nucleophilic strength. First, it is important to recognize that the two charged species,  and  are the two strongest nucleophiles. This is because the destabilizing negative charge present in these species may be neutralized by donating a lone pair to the formation of a chemical bond. As we know, opposite charges attract, so species bearing a full negative charge are drawn to electron-poor regions. Uncharged species such as water and ammonia carry a lone pair capable of bonding, but are less energetically drawn towards positive charges. 

Ammonia is a stronger nucleophile than water because nitrogen is less electronegative than oxygen. What this means is that the nitrogen-bound lone pair of ammonia is more loosely contained than the oxygen-bound lone pairs of water. As a result, they are more easily donated to form a bond at an electron-poor carbon.

From this trend, one might expect that fluoride ions would be less nucleophilic than chloride ions since fluorine is more electronegative. However, moving down a group of the periodic table, atomic radius increases. Anions are stabilized by spreading electron density across an electron cloud of greater volume, such as that of  compared to the smaller . As such, the correct ordering of species is II, I, III, IV.

In aprotic solvents, nucleophilicity increases with electronegativity when dealing with atoms in the same group (column on the periodic table).

When discussing nucleophilic strength, we can begin to see trends. However, it is important to note that the question asked for the strongest nucleophile in aprotic solvent. The correct answer is .

In polar protic solvents, electronegative nucleophiles tend to hydrogen bond with the solvent, inhibiting the nucleophile's nucleophilicity. However, in aprotic solvents, this does not occur and so basicity correlates to nucleophilicity.

Nucleophile are electron-rich species that form bonds with electron-poor species. When thinking in terms of acids and bases, bases tend to form bonds with protons making them strong nucleophiles while, acids usually donate protons making them weak nucleophiles.

In a protic solvent, the larger the atom the better the nucleophile. Atomic radius increases as you go down a group on the periodic table. Also, the more electronegative an atom/nucleophile, like fluorine, the higher the capability of forming hydrogen bonds with protic solvents, thus hindering their effectiveness as a nucleophile. 

The ordering from best nucleophile to worst nucleophile is as follows:

Smaller molecules are better nucleophiles than larger ones (they are not as sterically hindered).  is a better nucleophile than because nitrogen is less electronegative than oxygen (Look for the the lower electronegativity on the atom holding the lone pair of electrons). Thus, the lone pair of electrons on nitrogen are not as stable as those on oxygen, and will readily donate its pair of electrons to another species. 

The larger the atom, not molecule, the better the better the nucleophile . Atomic radius increases as you go down a group on the periodic table. Nucleophilicity can also be determined according to strength of the anion as a conjugate base. Remember that  is the strongest acid and thus has the weakest/least stable conjugate base.