A couple of examples of **non-admissible algebras**:

$\bullet$ It is known (see here, p.222) that a semisimple Hopf algebra is also separable (as an algebra). Consequently, **a semisimple but non-separable** algebra has to be non-admissible.

(An example of such an algebra could be the following: Consider a field $K$ and $L$ an inseparable extension. Then $L$ is a semisimple $K$-algebra but the algebra $L\otimes L$ is no longer semisimple).

$\bullet$ On the other hand, another example (outside the semisimple algebras) might be the case of **Weyl algebras and their deformations**. As far as I know, there is no known Hopf structure for the family of Weyl algebras. Furthermore, in an old paper of T.D. Palev, it is argued that it is impossible to even define a comultiplication on Weyl algebras. See: Is it possible to extend the deformed Weyl algebra $W_q(n)$ to a Hopf algebra? (the discussion of p.3-5 refers to the undeformed case).

**Edit:**

$\bullet$ I have recently found an interesting result leading to some other classes of examples of non-admissible algebras:

**Proposition:** Let $A$ be a connected graded algebra and let $x,y$ be non-zero elements in $A$ such that $xy=qyx$ for some $q\in k\backslash \{1\}$. Then there is no Hopf algebra structure on $A$.

Thus, $A$ is non-admissible in the sense of the OP's definition. This is proposition 2.7 from Connected (graded) hopf algebras. One of the authors appears to be the P.Gilmartin (who has provided the other answer above). So I am adding this reference and the following examples for the shake of completeness. As a consequence of the above proposition we get the following families of examples of non-admissible algebras (i am copying Corollary 2.8 of the formerly mentioned paper):

- Sklyanin algebras of any dimension
- skew-polynomial rings $k_{p_{i,j}}[x_1,x_2,...,x_n]$ (except for the case $p_{i,j}=1$ for all $i,j$)
- Quantum matrix algebras $\mathcal{O}_q(M_{n\times n})$ for $q\neq 1$
- non-commutative Koszul Artin-Schelter regular algebras of dimension $\leq 4$

$\bullet$ Using the above proposition (for $q=-1$), we can conclude that the **fermionic algebra** or the **algebra of anticommutation relations**, generated as an algebra by the fermion creation-annihilation operators $f_i^+, f_i^-$ for $i=1,2,...$ modulo the following relations
$$
\{f_i^-,f_j^+\}=\delta_{ij}, \ \ \{f_i^-,f_j^-\}=0, \ \ \{f_i^+,f_j^+\}=0
$$
(where $\{x,y\}=xy+yx$) for all $i,j=1,2,...$ is another example of non-admissible algebra i.e. of an algebra which does not admit a Hopf algebra structure. This might be of particular interest to mathematical physicists since this algebra describes the spin-$\frac{1}{2}$ particles.

(This last example is due to me and is not contained in the paper cited above. So if there is any disagreement ... put the blame on me!)