First: It is established in the Sunnah that there is encouragement to say this magnificent Dhikr, but without restriction to morning or evening.
Al-Bukhari (3293 and 6403) and Muslim (2691) narrated from the hadith of Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) said: "Whoever says: 'La ilaha illal-lah wahdahu la sharika lahu, lahul-mulk wa lahul-hamd wa huwa `ala kulli shay'in qadir' (There is none worthy of worship but Allah Alone, with no partner, His is the dominion and His is all praise, and He has power over all things) one hundred times in a day will get the same reward as given for freeing ten slaves. One hundred good deeds will be written for him, and one hundred sins will be erased from his account. It will be a protection from Satan for him that day until evening, and none will bring better than what he has brought except one who does more than that."
Imam An-Nawawi (may Allah have mercy on him) entitled a chapter in his book in Al-Adhkar (49): "A brief chapter concerning the virtue of Dhikr without time restrictions." And he mentioned this Hadith in it. Please see: (reference number 87).
An-Nawawi (may Allah have mercy on him) said: "The apparent meaning of the Hadith shows that one can attain this reward mentioned in this Hadith by saying this declaration of faith one hundred times in a day, whether consecutively or separately in different gatherings, or some at the beginning of the day and some at its end. However, it is better to say them consecutively at the beginning of the day so that it serves as a protection for him throughout his day." (Sharh Sahih Muslim, 17/17).
What Imam An-Nawawi (may Allah have mercy on him) established regarding the recommendation to say this in the beginning of the day was also mentioned by many other scholars and Hadith commentators.
Shaykh `Abd Al-`Aziz ibn Baz (may Allah have mercy on him) said: "It is prescribed for every Muslim man and woman to say in the morning of each day 'There is none worthy of worship but Allah Alone, without partner, His is the dominion and His is the praise, and He has power over all things' one hundred times, so that they will be protected from Satan that day until evening, based on what was previously mentioned in Al-Bukhari and Muslim from the Hadith of Abu Hurayrah (may Allah be pleased with him)..." (Al-Majmu` Al-Mufid Al-Mumtaz min Kutub Al-`Allamah Ibn Baz, p.145).
Moreover, this identical Dhikr also appears specifically with the restriction "when morning comes".. and "in the evening."
An-Nasa'i narrated in `Amal Al-Yawm wal-Laylah (p. 383), Ibn As-Sunni in `Amal Al-Yawm wal-Laylah (p. 65), and At-Tabarani in Ad-Du`a' (p. 125), through `Ubaydullah ibn Mu`adh, from his father, from Shu`bah, from Al-Hakam, from `Amr ibn Shu`ayb, from his father, from his grandfather, that the Messenger of Allah (peace and blessings of Allah be upon him) said: (Whoever says: "La ilaha illallah wahdahu la sharika lah, lahul-mulk wa lahul-hamd wa huwa `ala kulli shay'in qadir" [There is none worthy of worship but Allah alone, without partner, His is the dominion and His is the praise, and He has power over all things] one hundred times in the morning and one hundred times in the evening, no one will bring anything better except one who says more than that).
This chain of narration contains trustworthy narrators up to `Amr ibn Shu`ayb.
The manuscript of "`Amr ibn Shu`ayb from his father from his grandfather" does not fall below the rank of sound (Hasan) Hadith, if the narrator from him is trustworthy, as is the case with this chain of narration, especially in matters related to virtuous deeds whose fundamental basis in the Sunnah has been established.
Adh-Dhahabi (may Allah have mercy on him) said: "We are not among those who consider the manuscript of `Amr, from his father, from his grandfather, to be among the categories of authentic (Sahih) Hadith about which there is no dispute; due to the Wijadah method of transmission, and because it contains some objectionable narrations. Therefore, his Hadith should be carefully examined, what is objectionable from it should be avoided, and what remains should be narrated in the Sunan and rulings, considering its chain as sound. Indeed, great scholars have used it as evidence and generally deemed it reliable, while others hesitated slightly about it, but I do not know of anyone who completely rejected it." (Siyar A`lam An-Nubala', 5/175).
Al-Albani (may Allah have mercy on him) said: "This is a sound chain of narration, due to the well-known difference of opinion regarding `Amr ibn Shu`ayb from his father from his grandfather. That is why it was stated in Al-Fath (11/202): 'Its chain is authentic up to `Amr.'" (As-Silsilah As-Sahihah, 6/620).
Perhaps for this reason, many authors who wrote about Adhkar and daily acts of worship included it in "Morning and Evening Adhkar."
Al-Munthiri (may Allah have mercy on him) said in his book At-Targhib wat-Tarhib (1/447): "The Chapter on Encouraging the Recitation of verses and Adhkar in the Morning and Evening." He mentioned this Hadith in it (Please see: Hadith number 8 of the chapter).
Second: This Dhikr is also reported specifically for morning and evening remembrance, but with a different number: ten times.
Imam Ahmad reported in "Al-Musnad" (14/336) from Abu Hurayrah (may Allah be pleased with him) who said: The Messenger of Allah (peace and blessings of Allah be upon him) said: "Whoever says: 'There is none worthy of worship but Allah Alone, without partner, to Him belongs all sovereignty and praise, and He is over all things omnipotent,' ten times in the morning, will have one hundred good deeds written for him, one hundred bad deeds will be wiped away from him, and it will be equal to freeing one slave, and he will be protected by it that day until evening. Whoever says the same in the evening will have the same reward."
The verifiers of Al-Musnad said: "Its chain of narration meets the conditions of Al-Bukhari and Muslim." (End quote)
Similarly, Imam Ahmad in "Al-Musnad" (38/501-502) and Ibn Hibban (5/369) reported from Abu Ayyub (may Allah be pleased with him) who said: The Messenger of Allah (peace and blessings of Allah be upon him) said: "Whoever says in the morning: 'There is none worthy of worship but Allah Alone, without partner, to Him belongs all sovereignty and praise, and He is over all things omnipotent,' ten times, will have ten good deeds written for him, ten bad deeds will be erased from him, ten degrees will be raised for him, and it will be equal to freeing four slaves, and they will be a protection for him from Satan until evening. Whoever says them after praying Maghrib will have the same reward until morning.")
Shaykh Al-Albani classed it as sound and mentioned its various chains and supporting evidences in "As-Silsilah As-Sahihah" (6/134-137).
Third: His saying in the Hadith: "And none will come with anything better than what he has brought except a man who does more than that".
Abu Al-Walid Al-Baji (may Allah have mercy on him) said: "His saying: 'And none will come with anything better than what he has brought except one who does more than that' indicates that this is the pinnacle of remembrance of Allah the Exalted. It is rare for anyone to exceed it, which is why he said: 'And none will come with anything better than what he has brought'. If the saying did not carry this meaning, the statement would serve no purpose, since for any deed that a person does partially, naturally no one could bring better than what he brought except by bringing more than that." But this shows that this is the pinnacle in its category. Then he said: "except one who does more than that" to prevent the listener from thinking that exceeding this amount is forbidden, like repeating actions in Wudu.
And there is a second interpretation: it could mean that no one can bring anything better from all other forms of righteousness than what he brought, except a person who does more of this same type of deed. (Al-Muntaqa Sharh Al-Muwatta', 1/354).
Abu Al-`Abbas Al-Qurtubi (may Allah have mercy on him) said: "His saying 'except one who did more than that' means: one who said more. Thus, he classified speech as an action, as clearly stated in the other narration. Remembrance (Dhikr) is among the actions that only benefit when accompanied by intention and sincerity." (Al-Mufhim lima Ashkal min Talkhis Kitab Muslim, 7/21).
Al-Badr Al-Fayumi (may Allah have mercy on him) said: "This Hadith proves that if someone recites this declaration of Allah's oneness (Tahlil) more than one hundred times in a day, they would receive two rewards. First, they would get the reward mentioned in the Hadith for the hundred recitations. Second, they would receive additional reward for the extra recitations. This is not like the limits that we are forbidden to exceed, where increasing them brings no virtue or invalidates them. Examples of such limits include increasing the number of times in purification or the number of prayer units.
The phrase 'except one who does more than that' could have two meanings. First, it might refer to abundance in good deeds generally, not just in Tahlil specifically. Second, it could mean any increase, whether in Tahlil or other deeds. This second interpretation seems more apparent, and Allah knows best.
The apparent meaning of the hadith leaving it unrestricted indicates that this reward is attained by whoever says this Tahlil one hundred times in their day. This applies whether they say it consecutively or separately in different gatherings, or some in the beginning of the day and some at its end. However, it is better to say them consecutively at the beginning of the day, so it can serve as protection throughout the entire day. And Allah knows best." (Fat-h Al-Qarib Al-Mujib `ala At-Targhib wat-Tarhib, 4/339).
Shaykh `Abd Al-`Aziz ibn Baz (may Allah have mercy on him) said: "This indicates that there is no harm in adding more, whether one remembers Allah two hundred times or a thousand times; it is all good, with increased reward and blessings." (Majmu` Fatawa wa Maqalat Mutanawwi`ah by Ibn Baz, 26/92).
In conclusion: Saying "La ilaha illa Allah, wahdahu la sharika lah, lahul-mulk wa lahul-hamd, wa huwa `ala kulli shay'in qadir" (There is none worthy of worship but Allah alone, without partner, His is the dominion and His is the praise, and He has power over everything) has been established in the Sunnah to be said one hundred times daily. It is best to say it consecutively at the beginning of one's day. Whoever recites this Dhikr one hundred times after the morning prayer as part of the morning remembrances, and one hundred times after the `Asr prayer as part of the evening remembrances - this is good and prescribed. Whoever limits it to ten times in the morning and evening - this is also good, and its encouragement is established in the Sunnah. It should be recited one after another. However, if one is overcome by sleep, work, or similar matters, they should do what they can of what is prescribed and not abandon this Dhikr completely, due to its nobility and great significance. If someone does more than the prescribed amount, such as reciting it hundreds of times instead of a hundred, or tens instead of ten during their day - this is good and prescribed. The wording of the Hadith supports this, and the one who does this is upon good and is not an innovator.
And Allah knows best.