NP-hard problems with very fast exponential-time algorithms
The desired property holds for Independent Set (and probably other problems) in graphs of suitably bounded tree width.

Fix any constant $\epsilon&gt;0$ and consider the Independent Set problem restricted to graphs of tree width at most $n \log_2(1+\epsilon) = \Theta(\epsilon n)$, where $n$ is the number of vertices.  Call this problem $\Pi_\epsilon$.

Lemma 1. The problem $\Pi_\epsilon$ has an algorithm running in time $O((1+\epsilon)^n n^{O(1)})$.

Proof. This is a direct corollary of a result in [1] --- that Independent Set has an algorithm running in time $O(2^{\text{tw}(G)} n^{O(1)})$, where $\text{tw}(G)$ is the tree width of the given graph $G$.  $~~\Box$

Lemma 2. Assuming ETH, there is no algorithm for $\Pi_\epsilon$ that runs in time $2^{o(n)}$.

Proof. Per [2] Theorem 3.3, assuming ETH, there is no $2^{o(n)}$-time algorithm for Independent Set.  Suppose for contradiction that there is an algorithm $A$ for $\Pi_\epsilon$ running in time $2^{o(n)}$.  Construct an algorithm $B$ for Independent Set as follows:

Algorithm $B$ on input $G$, an arbitrary graph with $n$ vertices:


Let $G'$ be the graph obtained from $G$ by adding artificial vertices, each with no edges, to bring the total number of vertices to $n'=n/\log_2(1+\epsilon)$.  
Run the algorithm $A$ for $\Pi_\epsilon$ on $G'$ to find a maximum independent set $I'$ in $G'$.
Let $I$ be $I'$ minus all artificial vertices.  Return $I$.


Graph $G'$ has tree width at most $n'\log_2(1+\epsilon) = n$ because $G$ has tree width at most $n$.  So $A$ returns a maximum independent set $I'$ for $G'$ in time $2^{o(n')} = 2^{o(n)}$. $I'$ must consist of a maximum independent set $I$ in $G$ together with all the artificial vertices.  So $B$ returns a maximum independent set in $G$ in time $2^{o(n)}$, contradicting Theorem 3.3 of [2]. $~~~\Box$

Note: Edited 12/30/2019 to correct an error in the lower-bound argument.

[1] Invitation to Fixed-parameter Algorithms. Rolf Niedermeier. Oxford Lecture Series in Mathematics and its Applications, Vol. 31. Oxford University Press, Oxford

[2] Lower bounds based on the exponential time hypothesis. D Lokshtanov, D Marx, S Saurabh - Bulletin of the EATCS no 105, pp. 41 71, October 2011