Is there an elliptic curve with a bounded rank for all of its quadratic twists?
Consider the elliptic curve $E/\mathbb{Q}$ defined by: $y^2 = x^3 + ax + b$. Its quadratic twist, $E_D$, is given by $E_D: Dy^2 = x^3 + ax + b$
Is there a known instance of $E/\mathbb{Q}$ such that for every $D \in \mathbb{Z}$, the rank of $E_D/\mathbb{Q}$ is bounded above by a constant integer $M$?
In general these types of questions constitute a large collection of open problems. For a long time, people believed that ranks of elliptic curves over $\mathbb{Q}$ we unbounded. But modern heuristics tend to point in the other direction. A combination of the Goldfeld and parity conjectures would imply that the average rank of $E_D$ is $\frac{1}{2}$, so "on average" half the time the rank is $0$ and half the time it is $1$. Obviously this is overly simplistic since there are known lower bounds for quadratic twists having rank $\geq k$: see for example page 3 of this article by Silverberg.
The closest result I know relating to what you are asking about is this project of Watkins, et. al. In it they study the ranks of quadratic twists of the curve $y^2 = x^3-x$. Based on numerical evidence and a known modern heuristic, they conjecture that $$rank(E_D/\mathbb{Q}) \leq 7$$ for all $D$.