[lambertsimnel]

I think if you're a member of the public trading on an exchange, all you can do is ask an authorised entity to accept the "bid" or "ask" price (mentioned by brwnll above), which are set by other authorised entities. Maybe you can trade at another price by not using the exchange, in which case the market price would not reflect that trade, but I imagine that it would typically be difficult to find someone to trade with unless (maybe even if) you set a generous price.

I don't know, but maybe donating securities to charity and accepting a takeover bid could be considered examples of off-exchange trading.

If an authorised entity wanted to set an unusual "bid" or "ask" price, I think that would either trigger a trade (if it made the "bid" higher than the "ask") or (otherwise, assuming other authorised entities don't follow suit) merely signal that particular entity's unwillingness to trade that security at that the prevailing market price and have no effect on the market price.

I don't know whether this differs from exchange to exchange.