What are some of the more easily manipulated residues on an enzyme?
I'm assuming the OP concerns chemical means of modifying residues. Multiple protein residues other than Lysine may be labeled with an adduct. Here are some examples:

Histidine residues may be labeled using an epoxide attached to a ligand. (Chen et al., 2003)

Cysteine residues may be labeled with a nitrile derivative where the sulhydryl acts as the nucleophile. (Obella et al., 2007)

Arginine residues may labeled using a derivative of the naturally occurring methylglyoxal. (Oya et al., 1999)

Carboxylate functionality such as in Aspartate and Glutamate residues as well as the C-terminus may be labeled via the formation of an amide linker using a carbodiimide such as N-ethyl-3-N',N'-dimethylaminopropylcarbodiimide (EDC). (Gilles et al., 1999)


The above list of residues is not comprehensive, nor are the labeling agents. I recommend checking out the reviews by Basle et al. (2010) and Chen and Wu (2016).
References
Basle, E, Joubert, N, Pucheault, M (2010) &quot;Protein Chemical Modification on Endogenous Amino Acids&quot; Chemistry &amp; Biology 17:213-227. DOI: 10.1016/j.chembiol.2010.02.008
Chen, X and Wu, Y-W (2016) &quot;Selective chemical labeling of proteins&quot; Org. Biomol. Chem. 14:5417-5439. DOI: 10.1039/C6OB00126B
Gilles, MA, Hudson, AQ, Borders, CL (1990) &quot;Stability of water-soluble carbodiimides in aqueous solution&quot; Anal. Biochem. 184:244-248. DOI: 10.1016/0003-2697(90)90675-Y
Gong Chen, Alexander Heim, Doris Riether, Dominic Yee, Yelena Milgrom, Mary Ann Gawinowicz, and Dalibor Sames (2003) &quot;Reactivity of Functional Groups on the Protein Surface: Development of Epoxide Probes for Protein Labeling&quot; J. Am. Chem. Soc. 125:8130–8133. DOI: 10.1021/ja034287m
Oballa, RM, Truchon, JF, Bayly, CI, Chauret, N, Day, S, Crane, S, et al. (2007) &quot;A generally applicable method for assessing the electrophilicity and reactivity of diverse nitrile-containing compounds&quot; Bioorg. Med. Chem. Lett. 17:998–1002. DOI: 10.1016/j.bmcl.2006.11.044
Oya, T, Hattori, N, Mizuno, Y, Miyata, S, Maeda, S, Osawa, T, Uchida, K (1999) &quot;Methylglyoxal modification of protein. Chemical and immunochemical characterization of methylglyoxal-arginine adducts&quot; J. Biol. Chem. 274:18492-18502. DOI: 10.1074/jbc.274.26.18492