Researchers at UC Berkeley studied this exact question. About 40% of extensions contain at least one injection vulnerability. Some really popular extensions were vulnerable -- even a couple Google authored ones.
Starting with Chrome 18, extensions will be subject to
a CSP that enforces some of these bans [13]. Our study
partially motivated their decision to adopt the bans [1],
although the policy that they adopted is slightly stricter
than our recommendations. The mandatory policy in
Chrome 18 will ban HTTP scripts in core extensions, inline scripts, and dynamic code generation. Due to technical limitations, they are not adopting a ban on adding
HTTP scripts to HTTPS websites. The policy will remove all of the core extension vulnerabilities that we
found. The only extensions that the policy will permanently break are the two extensions that rely on eval.
The paper is mistaken in that these changes are actually coming with the manifest_version=2 property that is optional starting with Chrome 18, not required. However, we will be slowly transitioning the ecosystem over to this version and will eventually require it.
http://www.eecs.berkeley.edu/~afelt/extensionvulnerabilities...