How do I keep a player in the game when their character dies, without resurrecting that character?
Don't kill the players darlings
First, always make sure that the players are okay with losing their characters. Before running the game, explain that there might be character death and that they should create their characters accordingly.
I seldom run games where character death is a thing, but when I do I make sure the players know about it and are prepared for it before we start.
Bring a backup
Have a few backup characters in tow. Let the players create them so that they're invested in them and let them be mostly passive during play. This allows you to have new characters at-the-ready.
In a game I'm preparing now, the players will each create one "officer" of sorts and one "second in command". If their character dies, their replacement steps in. This also gives a reason for the secondary characters to be more passive.
One problem with this approach is party balance. The death of a character may not be a problem for the players, but the party looses in overall strength. This could, of course, also be an interesting challenge for them...
The chance encounter
A wild adventurer appears! Adventurer has joined your party!
Yes, it's a common target for ridicule and parody, but sometimes it just works. If the adventure is a bit cliché and campy anyway, this is a perfect fit. And a bit if sillyness is better than benching a player, right? Still, this is often not the best option so only use it when appropriate.
The prisoner
A more plausible reason for someone to be in the dungeon alone is to be the last surviving member of a previous, less successful, party of adventurers. Sure, the player will have to find some new gear, but that shouldn't be too hard. Hey, that other person who just died seems to have the same size as the newbie!
I've also used variations of this to introduce new players to an adventure or campaign.
The chance necromancer
A neutral necromancer lives in the dungeon. Who knew? She can resurrect the fallen character for a small fee. Or perhaps she's just imprisoned as well and will accept freedom as payment.
This option has to be done carefully. An undead PC can work fine as long as the player has fun with the character. Don't make the zombie too zombie-esque, let it be able to speak and retain intelligence. To make it feel a bit more like a proper death and not an upgrade, let there be a restriction to the resurrection so that the zombie can never leave the dungeon without losing its intelligence.
The reformed minion
After falling into a deadly trap, the characters are rescued by a minion of the dungeon. Before letting them out, the minion explains that he wants to change his evil ways and join the party. This can backfire if the other players decide not to trust the former minion, but it can also be a nice way to introduce a character with an instant backstory.
I've used variants of this to introduce new players and as long as the other players play nice it's a great way of fleshing out a group with some new blood.
The unreformed general
Give the player a major character in the dungeon. Perhaps even the main villain. Or simply invent a commander. Let the player look at the full map, plan a defence and command troops. This can become really engaging and rewarding for the player.
The downside is again that the party will become unbalanced. Although it might also help the rest of the party work together to keep each other alive in order to not gain another maniacal enemy...
The talking artifact
Want a weird twist to the adventure? Let the player take on the part of a talking artifact, such as an enchanted sword or shield. Try to make sure the thing is too big to shove into a backpack but also too valuable to leave behind.
I've never tried it myself, but the more I think about it, the more I want to.
UPDATE: Adventurers League changed the "+1 rule" in 2021.
An announcement from Q1 2021 gives the details. Basically, Adventurers League players may now simultaneously use any and all of the following sources when choosing player options like races, classes, subclasses, feats, spells, and backgrounds:
Other source books may also be available depending on the specifics of the campaign. The currently active campaigns (as of June 7, 2021) draw from subsets of the following list:
Note that variants and optional rules must be explicitly allowed, so the variant Human, Half-Elf, and Tiefling races in these books may or may not be available. The same goes for the customized origins in Tasha's. The best thing to do is to find and follow the character creation guidelines in the Adventurers League Player's Guide specific to your campaign, which can be found here.
The revised rules have been implemented in these active guides:
However, the active Seasonal Campaign Player's Guide (Plague of Ancients) remains the version from September 15, 2020. It's likely to be updated when the next Seasonal Campaign begins.
The active Guidelines for Using TCoE in Adventurers League Play also predate the change, so must be revised.
Goliaths
Regarding the original question, Volo's Guide to Monsters is on the list of source books available for all Adventurers League campaigns. Because goliaths are described therein, they're allowed in Adventurers League play.
However, under the updated rules, choosing a goliath no longer prevents the use of character options from other allowed source books like Xanathar's Guide to Everything.
Best Answer
In Adventurers League play, DM's can receive rewards for the time that they spend DM'ing, that they can use for their own PCs:
So, DM Quests, can be awarded by event organizers, as an incentive and a reward for the DM's who are giving up their time.
One of those quest categories is called: Giving DM and it's awarded for having 'Run a game as part of a charity event'.
So, what does that mean?
A DM not being paid, does not make the game a charity game.
It is fair to assume that the majority of the time, the majority of DMs are not being paid for their work. They are doing it voluntarily.
This is the reason that the DM Quests exist in general - to help incentivise (and reward) people to give up their own time and DM for others for no personal gain. So, not being paid does not make any given game a 'charity event'.
What is a 'charity event' then?
Well there we can refer to the general understanding of this term - rather than a DnD specific understanding:
And this seems to be the way that the DM Quests are using the term. Evidence of this can be found in the section of the Quest Sheet that details rewards:
Extra Life is a specific charitable effort, that is frequently supported by RPG players, which has raised more than $40 million for sick and injured kids since 2008.
So, a charity event, is an event (and in this context, one at which DnD occurs) which is specifically aimed at raising money for a charitable organisation or cause.
How could you get involved in a charity event?
If, as you say, no one in your country is running charitable events that you could volunteer to help with, then getting this DM Quest reward may prove tricky. However, given some time and effort, you might be able to organise your own event to raise money for a local organistation, or cause. Or, there might be charity events that you could support, and volunteer your time for, online.