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Generally, for the addition of dry HX to an alkene, it follows Markovnikov's rule. The hydrogen atom adds to the carbon with the most hydrogen atoms.

However, in the case of an alkene which does not have any hydrogen atom attached to its carbon, such as 2,3-dimethyl2-butene, would the electrophilic addition still take place?

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Electrophilic addition would likely still occur. Remember that Markovnikov's rule is an observational heuristic for "in electrophilic addition reactions of alkenes in which carbocations form, the more/most stable carbocation forms preferentially, and thus the product originating from that cation is the major product".

In this case, a carbocation can form. Nothing about the tetrasubstituted pattern prevents addition. The molecule is still planar and can be aproached by the electrophile. The methyl groups are electron donating by induction. The methyl groups 1) increase the electron density of the alkene and make it more nucleophilic, and 2) stabilize the carbocation intermediate.

Thus, 2,3-dimethyl-2-butene is likely more reactive than other alkenes toward electrophiles. For example:

$$\ce{(CH3)2C=C(CH3)2 + HBr -> (CH3)2BrC-CH(CH3)2}$$

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