What is the highest known lower bound for mate in N from starting position? Edit: It seems my question was not clear enough. Let me rephrase: What is the largest N for which we can knowingly say "chess, from the starting position, is not a forced mate in N moves"?

Chess is not solved, i.e. it is not known what the outcome from starting position is given perfect play.

However, if the starting position is win for either player, it is a mate in N for some N. Also by example, if we know for certain that the starting position is not winnable in 5 moves (for either player), 5 is a lower bound for N.

Roughly how deep is it feasible to search exhaustively from the starting position in practice? How high a lower bound for N is known?
This is essentially the question of what is the game complexity of chess. Note that by finiteness, we know that chess is determined, but we do not know if the starting position is a win for white, a win for black, or a draw.  The game complexity of chess is roughly the minimum number of positions we need to check in the game tree to determine the state of the initial position.  This is known as the Shannon number.  In the influential paper Programming a Computer for Playing Chess, Shannon estimated that the Shannon number is at least 10^{120).  Note that the number of particles in the Universe is estimated to be 10^(80).  To answer the question, we actually want to know the height of the game tree  when the initial position becomes determined.  We should also divide this height by 2, since a move in chess is typically considered a white and black move.   The branching factor of the tree is estimated to be about 30.  Thus, we can take the largest N such that 30^(2N) &lt; 10^(120).  

Answer. By the back of the envelope, N=40 works.  Coincidentally, this happens to be the length of an average game between grandmasters (although they often resign and do not actually play the game to conclusion).   

Edit. The moral of the story is that I was trying to estimate an upper bound for your lower bound.  The first part of Shannon's reasoning is not circular; he says that there are about 30 legal moves from each position, and this number is reasonably constant for the first portion of a game.  

Thus, we can estimate the current known value of N (which is really what you are asking, let's call this N') to be at most log_30 (C) where C is equal to the amount of computing power that has existed in the history of mankind.  Even with conservative estimates for C, we get something like  N' at most 20.  In practice, I don't think anyone has carried out this computation very far up the tree, since a priori we know that the computation becomes infeasible after a very short height and it is not necessary to exhaustively search the tree to write good chess programs.  

Note however, that you are asking a slightly weaker question, since it is possible that the initial state of the game is a draw with optimal play.  So, one could get bounds for N by writing a program whose goal was to not lose for as long as possible.  We could then play this program against the best programs or human players in the world and see what the length of a shortest game is.  Again, this does not properly answer the question, since we cannot assume that our opponents are playing optimally.  True optimal play requires full knowledge of the game tree, but we have seen that this is computationally infeasible.  Thus, the best we can currently do is approximate an optimally playing opponent with a Kasparov or a very good chess program.