Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion.
Now every Hopf algebra $H$ admits a one-dimensional comodule, namely the trivial comodule for a field $\mathbb{K}$ where
$$
\delta: \mathbb{K} \to \mathbb{K} \otimes H, ~~~~ k \mapsto k \otimes 1_H.
$$
Is there a name for a Hopf algebra that admits no one-dimensional comodule other than the trivial comodule?
 A: Q: Is there a name for a Hopf algebra that admits no one-dimensional comodule other than the trivial comodule?A: Not in the literature, but if you would like to coin a specific name for such a Hopf algebra, at the opposite extreme of pointed, you could call it blunt.
A class of blunt Hopf algebras is formed by the
Hopf algebras with free fusion semirings, see remark 1.3 in arXiv:1212.4763. Section 1.4 contains several examples of compact quantum groups associated with these Hopf algebras.
A: I am not really sure if this what the OP is looking for but i guess that a closely relevant notion here is that of connected Hopf algebras (i.e HAs which are connected as coalgebras). These are Hopf algebras which have no simple subcoalgebras other than $k\cdot 1_H$.
Let me attempt to explain that: The definition of pointed HAs can be stated equivalently as: HAs for which all simple subcoalgebras are 1-dim. This is due to the following well-known fact:

There is a 1-1 correspondence between the set of isomorphism classes of simple right $H$-comodules and the set of simple subcoalgebras of $H$.

(See: Indecomposable coalgebras, simple comodules, and pointed Hopf algebras, S. Montgomery, Proc. Amer. Math. Soc. 123 (1995), 2343-2351)
So, if you have a unique one-dimensional comodule (the trivial one) then under the above correspondence the unique subcoalgebra is $k\cdot 1_H$. But this is the definition of the connected HAs (Connected coalgebras essentially are: pointed, irreducible coalgebras. At the level of HAs,  connected HAs are the same thing as irreducible HAs. Universal enveloping algebras and most of their deformations are examples of connected HAs).
Edit: At a second read, and taking into account the way the OP is stated, maybe the notion of connected is more restrictive than desired: in the sense that for connected HAs, the trivial one-dim comodule is the unique simple comodule; while the OP asks for it to be the unique 1-dim comodule (leaving thus open the possibility that higher dim simple comodules may exist). But if this is the intention of the OP then i do not think -modulo my knowledge of course- that such a characterization exists.
A: Unless I'm mistaken you can say "coalgebra with conilpotent coabelianization" (the Hopf algebra structure doesn't seem relevant here). Namely, first note that any one-dimensional representation of a coalgebra facors through the coabelianization $H^{ab}$ (the maximal commutative sub-coalgebra) and a commutative coalgebra with a unique one-dimensional corepresentation is called conilpotent. This is equivalent to asking that the dual commutative ring $R = (H^{ab})^\vee$ viewed as a topological ring has a closed topologically nilpotent ideal $I$ with quotient $\mathbb{K}$.
