Company Names, addresses, dates:
  Dupuy, Rosser & Jones, cor Sycamore & Back Sts, Petersburg VA (1839-ca.1842) ^1,2
  Rosser & Anderson, Petersburg VA (ca.1843-ca.1848) ^3,4
  T.H. Rosser, Petersburg VA (ca.1848-ca.1858) ^5,6

Notes:
Thomas Henry Rosser (1818-1897)⁷ was one of the earliest drugists in Petersburg. He was a partner in the drugstore "Dupuy, Rosser & Jones", that started in 1839. The partners were William J. Dupuy and George W. Jones.¹
Rosser would have been about 20 years old at that time.
I couln't find any indication that Dupuy or Jones were Petersburg druggists before or after they had teamed up with Rosser. I also could not find any family connection between William J. Dupuy and P. E. DuPuy, who had a drugstore in Richmond.

The last newspaper ad I found for "Dupuy, Rosser & Jones" was in December of 1842², and in February of 1843, I found an ad for "Rosser & Anderson"³, so apparently Rosser found a new partner named Anderson. I could not find any mention where that store was located, other than it was in Petersburg.

1848 was the year when I found the last newspaper mention of "Rosser & Anderson", and the first mention of "T.H. Rosser" by himself.^4,5
Rosser was usually mentioned in advertisements for various concoctions which listed the drugstores where they were sold.
Then in 1856, I see the first mention of a concoction that Rosser produced, in an ad for a "Concentrated Essence of Bay Leaves" that was described as "unsurpassed in arresting the decay of the Hair", and not to be sold without the signiture of "Thos. H. Rosser, Manufacturing Chemist, Petersburg, Va."⁸
I found ads from 1857 that refer to it as "Rosser's Concentrated Essence of Bay Leaves", sold in Lynchburg. The ads describe the product as "Prepared by Thos. H. Rosser at his Chemical Laboratory, Petersburg, Va., and for sale by Druggists generally".⁹

By 1860, he had moved to Missouri, where he was a druggist.¹⁰

While he was a druggist, Rosser also had a career in the Army, where he was a Colonel.
In 1856, Rosser was active in organizing Virginians to temporarily move to Kansas. ¹¹
That was during the time that Kansas was in the process of becoming a state, and needed to decide whether or not to allow slavery. The Virginians were moving there so-as to vote in favor of slavery. Rosser was also involved in some of the battling that went on over the issue.

This is an 8" tall aqua bottle with a glass pontil scar on the base. It's embossed: square slutplate: T.H. ROSSER / DRUGGIST / PETERSBURG base: (glass pontil scar)