I am not sure what you expect as a result. Here is a transformation using a replacement rule:
1/Sqrt[1 + 1/t] /. (1/Sqrt[a_] :> Sqrt[Together[1/a]])
You may also try PowerExpand
, which is risky:
Simplify[PowerExpand@Together[1/Sqrt[1 + 1/t]],
t < 0 || t > 0]
The result does not seem particularly "simpler" than the original expression.