Of all our “a-” words, our very last is one of the mysteries.  The “a-” is not likely to be the intensifier preposition 1, but its true root is hard to disentangle from the distinct verb “awaken” or the obscure word of parallel meaning, “awecche.”

• 07.108 It was full morning when he awoke.
• 08.065 and when he was awake he was not pleased at all.
• 14.041 for they thought that war was awakening again,

“awake, v.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

“† aˈwecche, v.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

In this example of “a-” preposition 1, Bilbo is definitely not in the position of striding across his barrel.

• 09.056 even if he had managed to get astride his barrel,

“astride, adv., prep., and adj.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

This is mostly likely to be “a-” preposition 1, in the situation of straying.  In this case, it is said of good plans.

• 04.003 and of good friends like Elrond go astray sometimes

“astray, adv.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

This use of “a-” preposition 1 describes camp as being in the situation of stirring.

• 17.035 in the camp all was now astir,

“astir, adv.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

Thorin and Company only move ashore – “a-” preposition 1, in a situation – during the barrel adventure.

• 09.059 close ashore at one point
• 09.062 and waded ashore,
• 09.066 If they had come ashore
• 10.012 and helped ashore

“ashore, adv.2.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

Such an everyday thing as annoyance is not seen from the moment the dwarves enter the barrels until Bilbo is safely home in Bag-End.  This is a case of “an-” prefix 1, a particle handy for forming verbs, adverbs, and prepositions.

• 01.058 and annoyed.
• 01.142 (which annoyed Bilbo very much),
• 02.034 was most annoying.
• 02.108 and very annoyed:
• 03.018 or get annoyed with them.
• 05.098 But Bilbo was annoyed at the delay.
• 07.018 and you must be careful not to annoy him,
• 08.049 We are quite annoyed enough with you as it is.
• 09.034 and also rather annoyed.
• 19.035 as Bilbo noticed with annoyance.

“annoy, v.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

The “a-” prefix 1 intensifer is at work here, putting one intensely in the power of the muses.  OED notes that Shakespeare never used the word.  I find it notable that amusements end well before the capture in Mirkwood.

• 01.020 Very amusing for me,
• 04.013 and set dancing up by the roof to amuse them.
• 06.071 and they soon had a plan which seemed to them most amusing.

“amuse, v.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

While OED does not specify which “a-” this is, it does tell us it’s from “on middan”, so my money is on preposition 1 (in the situation).

• 03.050 Now they rode away amid songs of farewell
• 10.037 amid scenes of astonishing enthusiasm.
• 14.012 Amid shrieks
• 14.043 amid the ruined piles of the old town.

“amid, adv. and prep.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.

I am fascinated to notice that in Chapter 17, amazement only occurs in companionship with confusion.  This is a use of “a-” prefix 1 – towards and therefore intensifier of the sensation of being in a maze.

• 16.036 in amazement.
• 17.009 with amazement and confusion.
• 17.042 Amazement and confusion fell upon them all.

“amaze, v.” OED Online. Oxford University Press, March 2015. Web. 18 May 2015.