You can set a badge on your browser action with setBadgeText:
chrome.browserAction.setBadgeText({text: "10+"}); // We have 10+ unread items.
Note that if the text you pass is longer than 4 characters, it will be truncated to the first 4 characters.
Edit: Mar 2021
According to the new Manifest V3, the APIs have been changed. According to the document:
In Manifest V3 the chrome.browserAction and chrome.pageAction APIs are consolidated into a single chrome.action API.
So, your code for adhering to the new API shall be:
chrome.action.setBadgeText({text: "10+"}); // We have 10+ unread items.