Can I contract the same STD again after treatment?

Yes, reinfections are possible.

There are effective treatments available to cure chlamydia, gonorrhea, trichomoniasis, and syphilis. However, these antibiotics do not repair any tissue damage, nor do they prevent reinfection upon exposure at a later date. This is why it is particularly important for your current sexual partner to be also tested and treated if required.

Hepatitis C can also often be eliminated with antiviral medications, but again the treatment does not repair any tissue damage or prevent future infections. 

Hepatitis B and HIV are two STDs that cannot be completely eliminated from the body. Antiviral medications for hepatitis B can suppress the replication of the virus, thereby slowing progression of cirrhosis and reducing the risk of liver cancer, but they generally do not cure an HBV infection, so must continue for life. Prevention is the better approach for hepatitis B, as there is a very effective and safe vaccination available.