It has to be this.
An 8th tied to a half.
Or, possibly, if the grouping is 2+3, split the dotted quarter into a tied quarter and 8th.
I know you asked for NOT a tie.
But that isn't how rhythmic notation works.
And inventing a personal system for something as simple as this isn't sensible.
Sorry!
music.stackexchange.com/users/17003/Laurence Payne
Well, I suppose this might be theoretically possible with some sort of combination of nested tuplet brackets, but then you'd just create a bar that would be incomprehensible to normal musicians.
Keep in mind that where you put ties is not arbitrary: ties are frequently used to represent the beat structure, even when it is possible to write a single note value rather than two tied notes.
Here, since it isn't possible to write a single note of five eighth-note duration using standard note values, it's all the more important to use the tie to indicate the beat structure.
That is, if you're in 5/4, the use of that time signature generally indicates 5 quarter note "beats" per bar.
For example, say you were writing a bar of 4/4 time with two notes that are 1.5 beats long and a final quarter note.
While you could write it as dotted quarter-dotted quarter-quarter, many composers might write it out as dotted quarter followed by eighth tied to a quarter, then a final quarter.
The reason is that showing the third (strong) beat in a 4/4 bar improves legibility.
That rhythm isn't perhaps complicated enough to require such a tied note, but more complex rhythms might make it very helpful.
For your rhythm, with four 16ths on the first beat, and a dotted quarter on beat 2, it's actually helpful to show an eighth note on the upbeat of 3 tied to a half note on beat 4.
The only time you'd want to show a long duration note occurring on an off-beat in a complex metric situation like this is if it's a regularly occurring pattern.
(And if you are commonly having long-duration notes fall on the "and of 3" in a 5/4 bar, maybe you don't actually have a 5/4 meter, but instead a 10/8 meter that is segmented (5+5)/8 and it would be better to notate it that way.)
music.stackexchange.com/users/63809/Athanasius
As it turns out, you just plain can't do this with only one note value.
In fact, in Are ties necessary? , the questioner cites this exact problem of five eighth notes as one possible reason why ties must be used in our notational system.
Which leads to the next obvious question: why are you seeking a solution that uses only one a single note?
In all meters, you want to show the organization of the beats within the measure.
As such, for the sake of notational clarity, you would likely want to show this rhythm as an eighth note tied into a half.
music.stackexchange.com/users/21766/Richard
An eighth-note tied to a half-note (or some other such tied combination) is the standard way to notate a single note with a duration of five eighth-notes.
When you say you want “one note, not […] tied variations”, are you perhaps confusing ties with slurs?
They’re visually similar, but conceptually distinct: when a slur/tie connects noteheads of different pitches, then it’s a slur and just means they should be articulated legato, but when it connects two noteheads of the same pitch, it’s a tie, and really does mean they represent a single note of the combined total duration (unless it’s combined with certain other articulation markings, e.g. staccato dots).
As Richard’s answer says, the choice between using a half note tied to an eighth, or an eighth tied to a half, or other possible combinations, should be determined by how they fit into the meter of the whole bar: the break between the two notes in the combination should line up with as high-level as possible a division of the meter.
music.stackexchange.com/users/2197/PLL
I agree with the other answers that a tie is probably the best choice, since the musician is probably counting in 5/4, so they should be able to see the beat.
It only requires one tie.
But if you want to emphasize the fact that the measure is being split exactly in half, you could make a 2-note half-note duplet of the 5/4 measure, though as said before, this is already notatable with ties, and I think it's less convenient to read or to comprehend.
music.stackexchange.com/users/61288/awe lotta
I propose you indicate as instruction to the piece: a whole note=5/8 everything will be clear and you can forget all the rest, as it is your advice about the interpretation of your notation:
music.stackexchange.com/users/56219/Albrecht Hügli
In a bar of 5/4 there are: 4 sixteenth notes (1 beat), a dotted quarter note (1.5 beats) which leaves me with representing  2.5 beats as one note (not a half note tied to an eighth note or other tied variations). So what single note should I write to represent 2.5 beats?
How to I represent 5 eighth-notes as one note?
music.stackexchange.com
<rhythm>
music.stackexchange.com/questions/93100
music.stackexchange.com/users/65097/Jeong Whoj
An eighth-note tied to a half-note (or some other such tied combination)  is  the standard way to notate a single note with a duration of five eighth-notes. They’re visually similar, but conceptually distinct: when a slur/tie connects noteheads of different pitches, then it’s a slur and just means they should be articulated legato, but when it connects two noteheads of the same pitch, it’s a tie, and really does mean they represent a single note of the combined total duration (unless it’s combined with certain other articulation markings, e.g. staccato dots).  However, if you want to emphasize the fact that the measure is being split exactly in half, you could make a 2-note half-note duplet of the 5/4 measure, though as said before, this is already notatable with ties, and I think it's less convenient to read or to comprehend. An eighth-note tied to a half-note (or some other such tied combination)  is  the standard way to notate a single note with a duration of five eighth-notes.  
An eighth-note tied to a half-note (or some other such tied combination)  is  the standard way to notate a single note with a duration of five eighth-notes.  They’re visually similar to slurs.