For fun, I don't think what I suggest below will happen, but I believe it shows a problem in how the 22nd Amendment was written.
Here is the 22nd Amendment, at least the important part, and I have highlighted the part of concern:
No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this Article shall not apply to any person holding the office of President when this Article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this Article becomes operative from holding the office of President or acting as President during the remainder of such term.
Next up is the 12th Amendment, which states, "But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States".
So, the VP must be eligible to be president. The 22nd, does NOT say that a person serving two terms cannot be president, just that they cannot be
elected.
So, could Harris name Obama to be VP?