The Rich, The Independent One, The Wealthy.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Ghaniyy (in Arabic: ٱلْغَنيُّ), He is The One who is self-sufficient and independent. He does not need anything as He is completely satisfied and free from dependence. Allah is The One whose wealth and riches all others depend on.
From the root ghayn-nun-ya (غ ن ي), which has the following classical Arabic connotations: to be free from wants or needs, to be self-sufficient, independent, to be able to do without help from others, to be content, satisfied to be rich, wealthy, flourishing.
The name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ Al-Ghani, the rich or wealthy. This is one of the names of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ from the Qur'an as it is mentioned 18 times. It is often explained in pairs with Al-Mughni (The Enricher) for its closeness in meaning. But just as a note to self, Al-Mughni is one of the names where there isn't a consensus as being accepted as an official name of Allah as it's not mentioned in the Qur'an.
The word ghni (غْنِيَ) is often said to imply rich or wealthy. In the Qur'an, "Allah has certainly heard the statement of those [Jews] who said, innal laaha faqeerunw wa nahnu aghniyaaa "Indeed, Allah is poor, while we are rich." (Qur'an 3:181). In a hadith qudsi, the Prophet Muhammad ﷺ related Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as saying, "[...] O My servants, were the first of you and the last of you, the human of you and the jinn of you to rise up in one place and make a request of Me, and were I to give everyone what he requested, that would not decrease what I have, any more that a needle decreases the sea if put into it." This is how vast Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ wealth is. If every creation, human and jinn, both past, and present, were to ask Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ their innermost desire, and He was to give it to you, it would deplete Him as much as if you were to dip a needle in the ocean and try and remove some water with it. This is how rich Al-Ghaniyy is. His ability to provide is non-diminishable.
Living the rich life:
Now, if you were to ask Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ to gift you with a rich life, what would that mean? One dimension may include monetary success. But if that's all you care about, you'll have missed the aim of what a truly rich life is. Imagine a person who's been given wealth by Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ but with it a short lifespan, health problems, minimal love and support from friends or family, with limited capacity to think and reason. Would you say they are living a rich life? Their wish will have been wasted as they haven't even considered the purpose of their existence, which is to glorify and worship Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. They'll go through their entire life aimlessly, and on the day of resurrection, they'll be lost.
This is to illustrate that when we refer to Al-Ghani as rich, we don't mean in one aspect. But rather, He is abundant in all respects (i.e., knowledge, power, influence, might, time, etc.) In this light, we can see why Al-Ghaniy is often translated as 'The Self-Sufficient.' He is without the need for anything as He has it all. He is independent and entirely satisfied.
Qaalal lazee indahoo ‘ilmum minal Kitaabi ana aateeka bihee qabla ai yartadda ilaika tarfuk; falammaa ra aahu mustaqirran ‘indahoo qaala haazaa min fadli Rabbee li yabluwaneee ‘a-ashkuru am akfuru wa man shakara fa innamaa yashkuru linafsihee wa man kafara fa inna Rabbee Ghaniyyun Kareem
"Said one who had knowledge from the Scripture, 'I will bring it to you before your glance returns to you.' And when [Solomon] saw it placed before him, he said, 'This is from the favor of my Lord to test me whether I will be grateful or ungrateful. And whoever is grateful – his gratitude is only for [the benefit of] himself. And whoever is ungrateful – then indeed, my Lord is Free of need and Generous.'" — (Qur'an 27:40)
Wa Rabbukal ghaniyyu zur rahmah; iny yasha’ yuz hibkum wa yastakhlif mim ba’dikum maa yashaaa’u kamaaa ansha akum min zurriyyati qawmin aakhareen
"And your Lord is the Free of need, the possessor of mercy. If He wills, he can do away with you and give succession after you to whomever He wills, just as He produced you from the descendants of another people." — (Qur'an 6:133)
The Promise of Iblees:
Sometimes we can be very unappreciative of the blessings Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ has given us. He's provided for us, and we are not aware. We are not satiated by the things we possess. This is the curse of shaitan to look for the next thing, then the next thing, and after that, there is something else for us to fixate on. We are only finished chasing when we have reached our graves. This was the promise of Iblees. He vowed to make man ungrateful after his refusal to bow down to Adam (as), "Then I will come to them from before them and from behind them and on their right and on their left, and You will not find most of them grateful [to You]." (Qur'an 7:17) In the prophetic traditions of Muhammad ﷺ, you will find the best example of behavior. He never praised food or drink. He would express gratitude towards the one who prepared it. Abu Sa'id al-Khudri told that when God’s messenger finished his food he said, "Praise be to God who has given us food and drink and made us Muslims."  and also, Anas said, "When the Prophet, may Allah bless him and grant him peace, went to bed, he said, 'Praise be to Allah who has given us food and drink, and given us enough and given us refuge. How many people have neither enough nor refuge!'"  This is what it means to be cognizant of Allah's blessings.
Nothing can stand on human ambition and ability alone, we may be deluded to think so, but all success comes from Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. Do we fail to recognize His blessings? He's given the ability to see, breathe, hear, think, reflect, and love. This name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ should instill humility in the believer's heart. It should cause us to think about Him and see the blessings He's put around us. Every day we have food to eat, and we wake up and have the opportunity to spend time with our beautiful children, spouse, friends, and/or family. This is all a blessing of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ.
Qawlum ma’roofunw wa maghfiratun khairum min sadaqatiny yatba’uhaaa azaa; wallaahu Ghaniyyun Haleem
"Kind speech and forgiveness are better than charity followed by injury. And Allah is Free of need and Forbearing." — (Qur'an 2:263)
We can call upon Al-Ghani (The Rich) and Al-Mughni (The Enricher) whenever you are in need. Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is unlimited, and we should never fear asking of Him because who else would you then ask? The balance is that we should be careful to not be ungrateful or unsatisfied with the gifts He has already given us. Be deliberate in your worldly desires. If it benefits you, then ask of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. But if you only want something out of greed, pride, or to rise in social rankings, we should think twice about its importance. What would be best for the believer is to live a life of detachment just as Prophet Muhammad ﷺ did. For the one who has no need for anything but Allah, سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is the richest among the people.
Ali (ra) narrated: that a Mukatib came to him and said: “Indeed I am not capable of my Kitabah so aid me.” He said: “Should I not teach you words that the Messenger of Allah (ﷺ) taught me? If you had a debt upon you similar to the mountain of Sir, Allah would fulfill it for you. He said: 
Allahummakfini bihalalika `an haramika, wa aghnini bi fadlika `amman siwaka
"O Allah, suffice me with Your lawful against Your prohibited, and make me independent of all those besides You"
This is a beautiful supplication to ask Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ to help aid us in debts. But more generally, it is also both du'a of gratitude and detachment. Asking Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ to make us independent, free from all wants. We don't necessarily need more than we currently have in order to be at peace. This applies in the realm of rizq to make our careers enough for us, so we don't fall into prohibited acts of usury, cheating, stealing, etc. It also applies to marriages. To make us sufficient in our relationships, not desiring anything outside that has been forbidden.
A look into how the Prophet ﷺ lived:
In one of the longer hadith narrations, Umar bin Al-Khattab was said to visit the Prophet Muhammad ﷺ in his house, and what he saw made him cry. "Why are you weeping?' Umar replied, "O Allah's Messenger ﷺ! Caesar and Khosrau are leading the life (i.e., Luxurious life) while you, Allah's Messenger ﷺ though you are, is living in destitute". The Prophet ﷺ then replied, 'Won't you be satisfied that they enjoy this world and we the Hereafter?'"  Umar in the hadith describes the poverty-stricken lifestyle of Prophet Muhammad ﷺ in the hadith. He entered the home of Prophet Muhammad ﷺ to find him lying on a mat made of palm tree leaves. There was nothing to separate Prophet Muhammad ﷺ from the mat. When he got off the mat, you could visibly see the branch marks it left on his body. When he saw these marks, Umar began to weep. Even as the economic conditions of the Muslims improved, the Prophet Muhammad ﷺ stayed humble. Even in poverty, he ﷺ was offered many bribes by Pagan leaders to abandon his campaign, and he always denied it because with Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ He was sufficient.