Obtain and procure refer to getting something, or gaining possession of it, through some effort or process. Obtain is the general word for acquiring something using effort or by request (obtained a ticket for the show; obtain permission). Procure is an everyday word, in business and governmental contexts, for acquiring necessary supplies and equipment. In more general usage, however, procure suggests making a special effort to get something that may be difficult to obtain (managed to procure concert tickets). Procure is frequently used of necessities that conditions have made difficult to get (procuring organs to save lives, procuring food during wartime) and of illegal or illegally-obtained items, such as drugs and weapons.
Exceed can refer to going over or beyond a quantifiable or acceptable limit (exceed the speed limit; exceed their ability) or, more positively, to doing better than was anticipated (exceed our expectations). The range of this last sense extends to going beyond the outermost limits of something—or what seemed the outermost (a technology that will exceed the limits of brain imaging as we know it), and here it overlaps with the meaning of transcend. To transcend suggests both going beyond and rising entirely above the limits of something. A great speech that transcends the occasion it was written for, will have more than historical interest in the future because its truths have a broader reach. Frequent objects of transcend include boundaries, race, politics, division.
Both words describe people and things that affect us with their powerfully impressive or imposing presence that is of the highest dignity and grandeur. Majestic can refer to size, scope, or beauty, and suggests an exalted dignity (a majestic cathedral), but it is the natural world that most frequently inspires writers’ use of this word: majestic mountains; a majestic view; majestic creatures. August, the less common word, is most often applied to institutions, their members, and the associated architectural structures (the august halls of the U.S. Supreme Court). An august institution, body, assembly, or publication, inspires respect or reverence for its lofty purpose and the distinction and eminence of its members.