"Grander ambitions" refers to having bigger or more ambitious goals or aspirations. It means wanting to achieve greater levels of success, reach higher achievements, or have larger dreams. Full definition
However grand their ambitions, no - one here successfully manages to escape the grips of circumstance, of being in the wrong place at the wrong time. (politics.co.uk)
Also, every main character is similar in the way that they strive towards grand ambitions. (metacritic.com)
Nursing grand ambitions of her own, Nina sets her sights at Trigorin, who is never one to turn down firm young flesh. (npr.org)