How can I make the stock browser stop asking which browser I want to use? I have a number of browsers installed on my phone.  I haven't set any as the default because some pages simply render better on some browsers than on others.

When I use the stock browser, every single time I click a link it asks me what browser I want to use.  I'd like it to get the hint that if I chose the stock browser for this current session, then it should open the links in the stock browser and stop asking me.

Is there any way for me to stop the stock browser from asking me this without making it the default?
No. Android works with intents. Opening a link is asking for an intent to open an url. If you have more than 1 browser and you haven't set a default, Android will ask you which one to choose.

I can imagine certain browsers not following this rule of starting an intent and just stay in their own program but that's all as the program decide to handle it.