My citizenship does not allow me visa-free entry to most of the countries. Due to the nature of my work, I have to travel to various places quite frequently, and often require visitor/business visa. Right now it is quite a huge obstacle for me that some countries (e.g. the US and UK) require me to leave my passport at the embassy when I apply for a new visa or extension, which often takes around a week just to get my visa stamped and my passport mailed back to me. This prevents me from traveling (and doing my work) for at least a week. I am wondering if there are any ways to avoid having to leave the passport in the embassy, or travel without a passport?
I see that one option is to get a second passport, however even that is not enough in extreme cases (e.g. right now I have to extend my US visa and apply for UK visa, and need to make a flight to Europe at the same time). I believe my country does not allow to issue a third and more passports.
I wonder if there are any ways to negotiate/pay for priority services (AFAIK it's not an option even with priority processing for the UK) with embassies/visa centers so that I can keep my passport during the processing and come in person to get visas stamped without delays?