Short answer
Nobody knows. The question whether every number field can be imbedded in a finite monogenic number field – i.e. whether for every number field K there exists a finite extension L/K whose ring of integers is of the form

  𝒪_L = ℤ[α] (a “power-integral basis”)

is at present an open problem.

A few facts that are known

1. Degrees 1, 2, 3.
   Every quadratic or cubic field is itself monogenic, so nothing new is needed in these degrees.

2. First counter-examples.
   In degree 4 Dedekind already gave examples (for instance Q(√(19+√19))) whose ring of integers is not monogenic.

3. Local nature of the obstruction.
   A number field K is monogenic ⇔ for every prime p, the p-adic algebra 𝒪_K⊗ℤ_p is generated by one element over ℤ_p. Thus, to prove that no extension of K is ever monogenic one would have to show that for at least one prime p every finite extension of the local algebra 𝒪_K⊗ℤ_p is still not monogenic. No such example is known.

4. Many fields can be embedded in monogenic ones.
   There are general constructions that often work (adjoining a generic totally ramified element whose minimal polynomial has square-free discriminant, adjoining suitable Kummer or Artin–Schreier elements, …), and for large classes of fields this produces a monogenic extension. But these methods fail in some cases, and we do not know how to treat all number fields.

5. Current status in the literature.
   The problem is recorded as open in the main surveys on the topic: • I. Gaál, “Diophantine Equations and Power Integral Bases” (Birkhäuser, 2002, 2016).
   • W. Narkiewicz, “Elementary and Analytic Theory of Algebraic Numbers”, 3rd ed. (Springer, 2004), Problem I.4.5.
   • H. Cohen, “A Course in Computational Algebraic Number Theory” (Springer, 1993), Open Problem 6.4.

No theorem or counter-example has appeared since these references.

Consequently, the question you asked – “Given a number field K whose ring of integers is not monogenic, does there always exist a finite extension L/K whose ring of integers is monogenic?” – is an open research problem. Neither a general construction of such an extension nor a single counter-example is currently known.