The colloquial Australian-English phrase (as) full as a goog means very drunk.
—Cf. the synonymous Australian-English phrases as drunk as Chloe and in the grip of the grape.

Of unknown origin, goog is an Australian-English slang word for an egg. Sidney John Baker (1912-1976) proposed the following explanation in The Australian Language (Sydney and London: Angus and Robertson Ltd., 1945):

goog, an egg (a word formed perhaps on the sense of gog, in goosgog, a gooseberry ¹; U.S. slang has googs, spectacles—in all these cases roundness is implied*).
* Whence, full as a goog, equivalent to full as a tick, drunk.

¹ Perhaps compare with the humorous American-English noun cackleberry, denoting a hen’s egg, and composed of cackle, denoting the raucous clucking cry given by a hen, especially after laying an egg, and of berry.

These are the earliest occurrences of the phrase (as) full as a goog that I have found, in chronological order:

1-: From The Townsville Daily Bulletin (Townsville, Queensland) of Thursday 24^th November 1927:

WHARTON DIVORCE CASE.
Third Day.
(Before Mr. Justice Douglas.)

Mrs Towner (Anne Harris), continuing her evidence, stated that […] Mrs Wharton referred to her husband, saying he was “As full as a “goog” when he came home and had had more after.

2-: From On the Track, by ‘Bill Bowyang’, published in The Northern Miner (Charters Towers, Queensland) of Wednesday 4^th March 1931:

“Do you see that prosperous-looking bloke walking past the store over there?” said the little unshaven chap seated beside me on the pub verandah. “Well, there’s no chance of him asking us in to have a drink.” We were both stone motherless broke, and although he [misprint for ‘we’?] had been sitting on that verandah for three long hours not one darned passer-by had asked us if we had a thirst. I looked in the direction my mate pointed, and saw a prosperous-looking bloke walking down the street. “Looks like a wowser ²,” I said, “and I’ll wager he’s been barracking for the cold tea crowd all his life.” My companion grinned. “Don’t you believe it,” he said. “I can remember the time when that bloke was a fair hog for his booze, but one night when he was as full as a goog he scalped a man, and it gave him such a fright that he swore off the grog for life.”

² Of unknown origin, the noun wowser designates a Puritanical enthusiast or fanatic, especially a fanatical or determined opponent of intoxicating drink (cf. teetotal). In The Australian Language (Sydney and London: Angus and Robertson Ltd., 1945), Sidney John Baker (1912-1976) put forward the following hypothesis:

There are good reasons to suspect that wowser has English dialectal antecedents*.
* For instance, wusser, “a bad person, a ne’er-do-well”, wissere, “a teacher”, whizz, “a fussy troublesome person”, waw, “to whine, grumble, complain”.

3-: From The Cumberland Argus and Fruitgrowers’ Advocate (Parramatta, New South Wales) of Thursday 31^st May 1934:

“FULL AS A GOOG”

“What was his condition regarding sobriety?”
A witness was asked this question at the Parramatta Police Court on Monday.
“He was full as a goog,” the witness replied. “He was drunk—positively drunk. So drunk that he didn’t know whether he was going or coming.”
Which seems to suggest that the gentleman in question was hardly sober.

4-: From The Cumberland Argus and Fruitgrowers’ Advocate (Parramatta, New South Wales) of Thursday 16^th April 1936:

Proceeded against at the last sitting of Ryde Court for driving a motor car whilst under the influence of liquor, James Patteson, 22, was fined £10 and had his drivers’ license suspended until the date of its expiry and twelve months thereafter. Defendant pleaded guilty.
Sergeant Laney told the Magistrate, […] when the defendant was informed that he was drunk, he added: “Yes, sergeant, I’m as full as a goog.”

5-: From Truth (Sydney, New South Wales) of Sunday 24^th January 1937:

Moree Grazier “Full As a Goog”
WHAT HE SAID TO DOCTOR
(From “Truth’s” Moree Rep.)

Arthur Morris Boulton, well-known Moree grazier and wool and skin dealer, caused a laugh at the Moree Police Court the other day when he appeared on a charge of driving under the influence of liquor, and was fined £10 with a gaol alternative.
Sergeant Mullens said that when he saw Boulton in his lorry in the main street of Moree, and asked Boulton if he had been drinking, defendant did not beat about the bush.
“I have been drinking since two o’clock with some of the boys,” he said, “and have had about fifty lagers.”
The sergeant told the court that Boulton asked if a doctor could be brought along to the police station to examine him, and Dr. Ronald J. Hunter, Government medical officer, arrived on the scene.
“The doctor asked Bolton [sic] to bend down and touch his toes,” said the sergeant, “but Boulton, in a pained voice said, ‘Now, look here, doc., you are asking me to do something that I cannot do even when I am sober.” (Laughter.)
According to the sergeant Dr. Hunter then looked at Bouton [sic] with a stern eye and said to him, “Boulton, you’re drunk.”
Without batting an eyelid, Boulton replied: “Yes, doc, I’m as full as a little goog.” (Laughter.)
The sergeant informed Mr. Cole (for defendant) that Boulton always carried his drink well, and was able on this occasion successfully to park his lorry in Heber-street.
A fine of £10 was imposed.

6-: The phrase (as) full as a goog means sated with food in the caption to the following cartoon by B. E. Minns, published in The Bulletin (Sydney, New South Wales) of Wednesday 26^th May 1937:

“Now what about a little washing-up, Jacky?”
“Couldn’t eat another thing, thanks, missus. Full as a goog already.”