Upvote:1
It very much depends on your residence and citizenship, and which visas you're applying for. So we can't write an answer for every single visa. I'll try to generalise.
Often it requires that you send off your passport. So unless you have more than one passport (possible for some countries), you're clearly not going to be able to apply for more than one.
However, others like Russia, Uzbekistan require letters of invitation, so you can apply for those letters.
Uzbekistan, once I got my LOI, I was able to just pick up the visa on the road - I had to specify in advance which embassy, but got it from Riga, Latvia.
Others (like Russia, although I've been told this may have changed), insist that you must apply from either your country of citizenship, or country of residency, if you live elsewhere.
Still others let you apply on arrival and take 5 minutes.
So unfortunately, you'll have to do the dog work and research it all. Each country is different, so go through a site like visahq.com, and work out 1) where you have to apply for, 2) whether you need to send off your passport or 3) whether you can just pick up the visa on the road, or on arrival.
Good luck! Always hop into the [chat] if you want a chat about visas people have had trouble with.